What is Differentiable: Definition and 284 Discussions

In calculus (a branch of mathematics), a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its domain. A differentiable function is smooth (the function is locally well approximated as a linear function at each interior point) and does not contain any break, angle, or cusp.
More generally, for x0 as an interior point in the domain of a function f, then f is said to be differentiable at x0 if and only if the derivative f ′(x0) exists. In other words, the graph of f has a non-vertical tangent line at the point (x0, f(x0)). The function f is also called locally linear at x0 as it is well approximated by a linear function near this point.

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  1. R

    Differentiable structures and diffeomorphisms

    The definition of having multiple differentiable structures is that given two atlases, {(U_i ,\phi_i)} and {(V_j,\psi_j)} (where the open sets are the first entry and the homeomorphisms to an open subset of Rn are the second entry), that the union {(U_i,V_j;\phi_i,\psi_j)} is not necessarily...
  2. L

    G(x) is twice differentiable where g(4)=12 and g(5)=18. g(6)=?

    Homework Statement Let g be a twice differentiable function with g'(x)>0 and g''(x)>0 for all real numbers x, such that g(4)=12 and g(5)=18. Of the following, which is a possible value for g(6)? a. 15 b. 18 c. 21 d. 24 e. 27 Answer: e. 27 Homework Equations The Attempt...
  3. H

    0 when irrational, 1/q in lowest terms with rational not differentiable.

    Homework Statement Hi, I have this function: f(x ) = 0 (x is irrational) or f(x) = 1/q for rational p/q in lowest terms. show that this function is not differentiable anywhere The Attempt at a Solution This is the answer from the solutions book: consider [(f(a+h) - f(a ) ) /...
  4. P

    (Complex) Analytic as opposed to Differentiable

    Good day! Well, I've just started this year with Complex Analysis (we're using "Complex Variables and Applications" 8th ed by Brown and Churchill) and as I'm going through some assignment questions, I noticed that one of the questions states: "Is g analytic at any point of C (as opposed to...
  5. C

    Convergence, differentiable, integrable, sequence of functions

    Homework Statement For k = 1,2,\ldots define f_k : \mathbb{R} \to \mathbb{R} by f_k(x) = \sqrt{k} x^k (1 - x). Does \{ f_k \} converge? In what sense? Is the limit integrable? Differentiable? Homework Equations The Attempt at a Solution I don't know how to approach this...
  6. N

    Prove differentiable for a function

    Homework Statement f(x,y) is differentiable in (0,0) and f(0,0)=0 q(t) devirate t=0 and q'(0)=1 q(0)=0 let it be g(x,y) = q(f(x,y)) prove that g differentiable in (0,0) and that f_{x}(0,0) = g_{x}(0,0)Homework Equations all calculus The Attempt at a Solution Well my idea is like this First I...
  7. M

    Differentiable function with g(0) = 0 and etc

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  8. M

    Composition of two differentiable functions

    Homework Statement Is the composition of two differentiable functions always differentiable? E.x. h(x) = sin(x) k(x) = 1/x for x not equal 0 Does this automatically mean h(k(x)) is differentiable? Thank you, M
  9. N

    Differentiating Composition of Smooth Functions

    Homework Statement Let f: M \rightarrow N , g:N \rightarrow K , and h = g \circ f : M \rightarrow K . Show that h_{*} = g_{*} \circ f_{*} . Proof: Let M, N and K be manifolds and f and g be C^\infinity functions. Let p \in M. For any u \in F^{\infinity}(g(f((p))) and any...
  10. H

    Differentiable multivariable functions

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  11. P

    How does classical mechanics change if motion was not infinitely differentiable?

    Many "theoretical mechanicians" seem to awesome that motion is a {C^\infty } function(at least that is how I learned it). However, it seems like the postulates of Newtonian/Lagrangian/Hamiltonian/Vakonomic mechanics seem to "work" in the general case where only the motion is a {C^2}(ie the...
  12. C

    Prove Finite Dimensional Normed Vector Space is Differentiable

    Homework Statement Let V be a finite dimensional normed vector space and let U= L(V)*, the set of invertible elements in L(V). Show, f:U-->U defined by f(T)= T-1 is differentiable at each T in U and moreover, Df(T)H = -T-1HT-1 where Df(T)= f'(T). Homework Equations Apparently...
  13. D

    Differentiable functions proof

    Homework Statement Consider a function g : (a, b)-->R. Assume that g is differentiable at some point c in (a,b) and that g'(c) is not = 0. Show that there is a delta > 0 so that g(x) is unequal to g(c) for all x in V_delta(c)\{c}intersect(a,b) Homework Equations The Attempt at a...
  14. R

    Differentiable manifold not riemannian

    I'm looking for a simple example of a differentiable manifold that doesn't have an associated riemann metric. thanks
  15. R

    Non 2nd-countable, Hausdorff, differentiable n-manifold?

    I am trying to find a Hausdorff topological space that is not second-countable but otherwise a DIFFERENTIABLE n-manifold. I can't figure it out. Does it exist? :smile: I read about the classical example of L=\omega_1\times[0,1) with lexicographical order and the order topology. It's Hausdorff...
  16. W

    Which single-valued differentiable function between 0 and 1 satisfies

    I look for a function u(x) with u(0)=0 and u(1)=1, which is single-valued and differentiable on the entire interval x= [0,1] and allows one to choose the derivatives u'(0) and u'(1) through two free parameters. Seems simple enough, right?
  17. N

    For what values of c and d will x be differentiable for all values

    For what values of c and d will x be differentiable for all values f(x)=\left\{\begin{array}{cc}cx+d,&\mbox{ if } x\leq -1\\cx^3+x+2d, & \mbox{ if } x>-1\end{array}\right. i took the derivative on each side f'(x) = c f'(x) = 3cx²+1 c = 3cx²+1 so i get c as -(1/2) but...
  18. L

    If f is differentiable, is f ' continuous?

    So, a certain discussion occurred in class today... If f is differentiable, is f ' continuous? At first sight, there seems no reason to think so. However, we couldn't think any counterexample. It also seems logical that f' is continuous since otherwise f wouldn't be differentiable. For...
  19. A

    What's the difference between analytic and continuously differentiable?

    What's the difference between "analytic" and "continuously differentiable?" I'm reading Gamelin's Complex Analysis book, and he talks about f(z) being analytic if it is continuously differentiable and satisfies the Cauchy-Riemann equations. But if f(z) is continuously differentiable, doesn't...
  20. L

    Bijective & continuous -> differentiable?

    Is a bijective continuous function:[a,b]->[f(a),f(b)] differentiable? I think it has to be. continuity between two distinct values of f(a) and f(b): it got to take all the values between f(a) and f(b) at x in [a,b], by the intermediate value theorem. if f is bijective, at [a,b], f(x) can't go...
  21. T

    Show f is differentiable but partial derivatives are not continuous

    Homework Statement Define f: Rn --------> R as f(x) = (||x||^2)*sin (1/||x||) for ||x|| ≠ 0 f(x) = 0 for ||x|| = 0 Show that f is differentiable everywhere but that the partial derivatives are not continuous. Homework Equations The Attempt at a Solution Showing that it is...
  22. T

    Differentiable function / showing a set is a neighbourhood of a point

    Homework Statement Suppose f:Rn ----> R is differentiable at the origin, (but not necessarily elsewhere), that f(0)=0, and that there is a constant c such that the norm of the gradient of f at zero is less than c. (||˅f(0)||<c ) Show that the set U = {x e Rn : ||f(x)||< c||x|| } is a...
  23. S

    Need assistance(Gussian curvature and differentiable vector fields)

    Need urgent assistance(Gussian curvature and differentiable vector fields) Hi I have a very difficult problem where I know some of the dots but can't connect them :( So therefore I hope that there is someone who can assist me (hopefully :)) Homework Statement Let S be a surface with...
  24. D

    Determining if a function is differentiable at the indicated point

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  25. B

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  26. A

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  27. S

    Is f totally differentiable at (0,0)?

    Homework Statement For any (x,y) other than (0,0). f(x,y)=\tan(x·y)\sin\left(\frac{1}{x^2+y^2}\right) For (x,y) = (0,0) f(x,y) = 0 Is f totally differentiable? Homework Equations The Attempt at a Solution If the function is not continuous, it can't be differentiable...
  28. J

    If a function f is differentiable at a point x = c of its domain, then

    If a function f is differentiable at a point x = c of its domain, then must it also be differentiable in some neighborhood of x = c?
  29. S

    Can Different Choices of w Lead to Continuously Differentiable u?

    Homework Statement Given: w=9(cos(\theta)+sin(\theta)) and u=r*w. 1) Is u continuously differentiable? 2)Is it possible to get u continuously differentiable with a different w? Homework Equations The Attempt at a Solution 1) u is continuously differentiable since w is in terms...
  30. T

    Limit of differentiable functions question

    f(x) and g(x) are differentiable on 0 f(0)=g(0)=0 \lim _{x->0}\frac{cos(f(x))-cos(g(x))}{x^2}=\lim _{x->0}\frac{-2sin(\frac{f(x)+g(x)}{2})sin(\frac{f(x)-g(x)}{2})}{x^2}=-2 because i can use (sin x)/x=1 here is it ok??
  31. B

    Is a graph Continuous and differentiable at a given point

    Homework Statement F f(x)={(2x-1)/Absolute value(2x-1) x cannot equal (1/2) { 0 x = (1/2) a) is f continuous at X = (1/2) explain b) is f differentiable at x = (1/2) explain Homework Equations I have made the graph and x is a point...
  32. Redbelly98

    F(x) differentiable at / near x=0?

    This came up in a recent discussion about l'Hopital's Rule. Suppose f(x) has a derivative at x=0, that is f'(0) exists. Is it necessarily true that f(x) is differentiable in some open interval containing x=0? Others--who know calculus better than I--say no, f(x) is not necessarily...
  33. I

    R twice continuously differentiable function proof

    r twice continuously differentiable function proof... Homework Statement Help :frown: if f:[a,b] \rightarrow R is twice continuously differentiable, and f(x)\geq 0 for all x in [a.b] and f ''(x) \leq 0 for all x in [a,b] prove that 1/2 (f(a) + f(b)) (b-a) \leq \int f(x)dx \leq(b-a)...
  34. T

    Prove that this function differentiable endles times

    prove that this function differentiable endles times on x=0 ?? http://img502.imageshack.us/img502/6778/83126617mm0.th.gif i was told "once we can express the function as a power series around zero and it is differentiable at zero, we know it is infinitely differentiable" differentiable...
  35. J

    A differentiable function whose derivative is not integrable

    Homework Statement Suppose g is a differentiable on [a,b] and f = g', then does there exist a function f which is not integrable? Homework Equations The Attempt at a Solution I've tried to look at pathological functions such as irrational, rational piecewise functions. but the...
  36. S

    Complex show differentiable only at z=0

    Homework Statement Show that f(z) = zRez is differentiable only at z=0, find f'(0) The Attempt at a Solution This should be easy. I find the limit as z_0 approaches 0 of [f(z+z_0) - f(z)]/(z_0) for this function...expand it out, simplify, and find what the limit is when z_0 is...
  37. J

    Proving Inequality for Continuously Differentiable Functions on Closed Interval

    I'm having trouble with this inequality: let f be (real valued) continuously differentiable on [0,1] with f(0)=0, prove that sup_{x\in[0,1]} \left|f(x)\right| \leq \int^{1}_{0}\left|f\acute{}(x)\right| dx Thanks for any help.
  38. K

    Show h is differentiable

    Homework Statement Define h(x)=x^3sin(1/x) for x\neq0. and h(0)=0. Show h is differentiable everywhere and that h is cont everywhere, but fails to have a derivative at one point. Homework Equations The Attempt at a Solution [h(x)-h(0)]/[x-0]=x^2sin(1/x) h is diff everywhere...
  39. K

    Continuous and differentiable Of Cos

    Homework Statement how could i prove that cos x= sum (n=1 to 00) [((-1)^n) * x^(2n)/((2n)!)] is continuous and differentiable at each x in R Homework Equations the Taylor Expansion of cosine is the given equation The Attempt at a Solution basically i need to prove that the...
  40. S

    Differentiability of Inverse Map in Bounded Linear Transformations

    Let B(V,V) be the set of bounded linear transformations from V to V. Let U be the set of invertible elements of B(V,V) and define the map ^{-1}: U\rightarrow U by ^{-1}(T)=T^{-1} Show that the map ^{-1} is differentiable at each T \in U.
  41. quasar987

    Question about differentiable structures

    If M is a topological manifold, a smooth structure A (or maximal atlas) on M is a set of smoothly compatible charts of M that is maximal in the sense that if we consider any chart that is not in A, then there is some chart in A with whom it is not smoothly compatible. Now, it is a fact that...
  42. I

    Differentiable and uniformly continuous?

    differentiable and uniformly continuous?? Homework Statement Suppose f:(a,b) -> R is differentiable and | f'(x) | <= M for all x in (a,b). Prove f is uniformly continuous on (a,b). Homework Equations The definition of uniform continuity is: for any e there is a d s.t. | x- Y | < d...
  43. L

    How to find whether this function is differentiable

    Homework Statement F(x)= {1/2x+1 when x=<2 {squareroot(2x) when x=>/=2 is it differentiable at x=2. Homework Equations (f(x)-f(2))/(x-2) The Attempt at a Solution So i know i ahve to take the limit from both the negative and positive of 2, and determine if they are equal. But...
  44. J

    What is the Equation for the Tangent Line of g(x) at Point x=2?

    Homework Statement Let f be a function differentiable function with f(2) = 3 and f'(2) = -5, and let g be the function defined by g(x) = xf(x). Which of the following is an equation of the line tangent to the graph of g at point where x=2? a. y=3x b y-3 = -5(x-2) c y-6 = -5(x-2) d y-3 =...
  45. JasonJo

    Tangent bundle of a differentiable manifold M even if M isn't orientable

    This is a problem many of the grad students have probably encountered, it's in Chapter 0 of Riemannian Geometry by Do Carmo. Do Carmo proved that the tangent bundle of a differentiable manifold is itself a differentiable manifold by constructing a differentiable structure on TM, where M is a...
  46. J

    Twice differentiable functions

    Hi all. Having a little trouble on this week's problem set. Perhaps one of you might be able to provide some insight. Homework Statement f:[a,b] \rightarrow \mathbb{R} is continuous and twice differentiable on (a,b). If f(a)=f(b)=0 and f(c) > 0 for some c \in (a,b) then \exists...
  47. M

    Proving No Differentiable f Such That f Circ f = g

    Suppose the real valued g is defined on \mathbb{R} and g'(x) < 0 for every real x. Prove there's no differentiable f: R \rightarrow R such that f \circ f = g.
  48. E

    About 1-1 differentiable functions

    Hi, I've been thinking about a problem in Spivak's Calculus on Manifolds and noticed that it can be proven quite cleanly if the following is true: Let g:R^n->R^n be a differentiable 1-1 function. Then we can find a point s.t. det g'(x) != 0. Geometrically this means that the best linear...
  49. S

    Non continuously differentiable but inner product finite

    Hello, I was trying to understand Green's function and I stumbled across the following statements which is confusing to me. I was referring to the following site http://www.math.ohio-state.edu/~gerlach/math/BVtypset/node79.html Here the author says the following "What if $ u$ is...
  50. M

    Differentiable Vector Functions

    Homework Statement Show through direct application of definition that the function f(x,y) = xy is differentiable at (1,1) The Attempt at a Solution I know that all functions of the class C1 are differentiable and that a function is of the class C1 if its partial derivatives...
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