# What is Differentiability: Definition and 196 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. ### A question regarding continuous function on a closed interval

##(f(c) - f(a))((f)(b) - f(c)) <0## tells us that there are two cases: ##f(c) >f(a), f(b) ## ##f(c) <f(a), f(b) ##. I guess we need to define a new function here that let us use the Rolle's theorem.. But it is not clear enough how to do so.

22. ### MHB Is Differentiability at the Origin Determined by Partial Derivatives?

Hey! :o Let $g:\mathbb{R}\rightarrow \mathbb{R}$ be arbitrary and $f:\mathbb{R}^2\rightarrow \mathbb{R}$ be defined by $f(x,y)=yg(x)$. I want to prove that $f$ is differentiable in the origin if and only if $g$ is continuous in $x=0$. So that $f$ is differentiable in $(0,0)$ does the...
23. ### MHB Differentiability of mappings from R^n to R^p .... .... D&K Lemma 2.2.3 .... ....

I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 2: Differentiation ... ... I need help with the proof of Lemma 2.2.3 ... ... Duistermaat and Kolk's Lemma 2.2.3 and its proof read as follows: I do not...
24. ### MHB Differentiability of mappings from R^n to R^p .... .... D&K Defn 2.2.2 ....

I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 2: Differentiation ... ... I need help with understanding an aspect of Definition 2.2.2 ... ... Duistermaat and Kolk's Definition 2.2.2 reads as...
25. ### MHB Differentiability of mappings from R to R .... ....

I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 2: Differentiation ... ... I need help with an aspect of the proof of Proposition 2.2.1 ... ... Duistermaat and Kolk's Proposition 2.2.1 and its proof...
26. ### MHB Infinite Differentiability and Analyticity.

I want to show that the function defined as follows: $f(x)=e^{-1/x^2}$ for $|x|>0$ and $f^{(k)}(0)=0$ for $k=0,1,2,\ldots$ is infinitely differentiable but not analytic at the point $x=0$. For infinite-differentiability I used the fact that $\lim_{|x|\to 0^+} x^{-n} e^{-1/x^2}=0$ for every $n$...
27. ### Differentiability and functional equations

Homework Statement Let f((x+y)/2)= {[f(x)+f(y)]/2} for all real x and y {f'(x)=first order derivative of f(x)} f'(0) exists and is equal to -1 and f(0)=1. Find f(2) Homework Equations Basic formula for differentiablilty: f'(x)=limit (h tends to 0+) {[f(x+h)-f(x)]/h} The Attempt at a...
28. ### AP Calculus BC: Differentiability and continuity

Homework Statement The function h is differentiable, and for all values of x, h(x)=h(2-x) Which of the following statements must be true? 1. Integral (from 0 to 2) h(x) dx >0 2. h'(1)=0 3.h'(0)=h'(2)=1 A. 1 only B.2 only C. 3 only D. 2 &3 only E. 1,2 &3 Homework Equations None that I am...
29. ### I Limits to directly check second order differentiability

Sorry, I mistakenly reported my own post last time. But later I realized that these limits do work. So, I'm posting this again. I'm using these limits to check second-order differentiability: $$\lim_{h\rightarrow 0}\frac{f(x+2h)-2f(x+h)+f(x)}{h^2}$$ And, \lim_{h\rightarrow...
30. ### MHB Find x for Differentiability of |x2-4x+3|

Hello all, I wish the find the values of x for which the following function is differentiable: $\left | x^{2}-4x+3 \right |$ I got the point that the function is continuous apart from x=1,3. I need to find if it is differentiable at x=1,3, using the limit definition of the derivative. I am...
31. ### I Differentiability of multivariable functions

What does it mean for a ##f(x,y)## to be differentiable at ##(a,b)##? Do I have to somehow show ##f(x,y)-f(a,b)-\nabla f(a,b)\cdot \left( x-a,y-b \right) =0 ##? To show the function is not though, it's enough to show, using the limit definition, that the partial derivative approaching in one...
32. ### Proof of differentiability for <x,x>

Homework Statement Hi everybody! I'm struggling to solve the following problem: Let ##< \cdot, \cdot >## be an inner product on the vector space ##X##, and ##|| \cdot ||## is the norm generated by the inner product. Prove that the function ##x \in X \mapsto ||x||^2 \in \mathbb{R}## is...
33. ### I Open interval (set) end points and differentiability.

When we talk about differentiability on a Set X, the set has to be open. And if a set X is open there exists epsilon> 0 where epsilon is in R. Then if x is in X, y=x+ or - epsilon and y is also in X But this contradicts to what i was taught in high school; end points are excluded in the open...
34. ### I Differentiability of convolution

If f and g are continuous functions on the right half-line, [0,∞], then f✶g, the convolution of f and g, is defined by f✶g(x) = ∫[0,x] f(t)g(x-t)dt. I would like to know if f✶g is a differentiable function of x. If, for example, g(t) = 1 for t ≥ 0 then f✶g(x) = ∫[0,x]f(t)dt has a derivative...
35. ### Differentiability of a function -- question on bounding

Homework Statement I need to see if the function defined as##f(x,y) = \left\{ \begin{array}{lr} \frac{xy^2}{x^2 + y^2} & (x,y)\neq{}(0,0)\\ 0 & (x,y)=(0,0) \end{array} \right.## is differentiable at (0,0) Homework Equations [/B] A function is differentiable at a...
36. ### Differentiability of piece-wise functions

Hello, Me and my friend were talking about differentiability of some piece-wise functions, but we thought of a problem that we could were not able to come to an agreement on. If the function is: y=sin(x) for x≠0 and y=x^2 for x=0, Is this function differentiable? The graph looks like a normal...
37. ### Showing a limit exists using differentiability

Homework Statement Assume f:(a,b)→ℝ is differentiable on (a,b) and that |f'(x)| < 1 for all x in (a,b). Let an be a sequence in (a,b) so that an→a. Show that the limit as n goes to infinity of f(an) exists. Homework Equations We've learned about the mean value theorem, and all of that fun...
38. ### Is ln(x) differentiable at negative x-axis

Since lnx is defined for positive x only shouldn't the derivative of lnx be 1/x, where x is positive. My books does not specify that x must be positive, so is lnx differentiable for all x?
39. ### Necessary and sufficient condition for differentiability

Alright, so now that I think have some more "mathematical maturity", I have decided to go back and review/re-learn multivariable calculus. I've just started, and have gotten to differentiation. From what I have seen, most books state the following sufficient condition for differentiability: A...
40. ### Continuity and Differentiability of f:R->R

Homework Statement Mod note: Edited the function definition below to reflect the OP's intent. Suppose f:R->R is continuous. Let λ be a positive real number, and assume that for every x in R and a>0,f(ax)=aλ f(x). (a) If λ > 1 show that f is differentiable at 0. (b) If 0 < λ < 1 show that f is...
41. ### Checking if f(x)=g(x)+h(x) is onto

This is picture taken from my textbook. I understood the last two statements "To check whether..". A function is one if its strictly increasing or decreasing. But I am not able to understand the first statement. Polynomials are continuous functions. Also, a continuous function ± discontinuous...
42. ### Finding the number of rational values a function can take

Homework Statement ##f(x)## is a continuous and differentiable function. ##f(x)## takes values of the form ##^+_-\sqrt{I}## whenever x=a or b, (where ##I## denotes whole numbers) ; otherwise ##f(x)## takes real values. Also, ##|f(a)|\le |f(b)|## and ##f(c)=-1.5##. Graph of ##y=f(x)f'(x)##: The...
43. ### Mean Value Theorem/Rolle's Theorem and differentiability

Homework Statement Let f(x) = 1 - x2/3. Show that f(-1) = f(1) but there is no number c in (-1,1) such that f'(c) = 0. Why does this not contradict Rolle's Theorem? Homework EquationsThe Attempt at a Solution f(x) = 1 - x2/3. f(-1) = 1 - 1 = 0 f(1) = 1 - 1 = 0 f' = 2/3 x -1/3. I don't...
44. ### Differentiability of the absolute value of a function

The derivative of ##|f(x)|## with respect to ##x## is ##f'(x)## for ##f(x) > 0## and ##-f'(x)## for ##f(x) < 0##. However, it is undefined wherever the value of the function is zero. I was wondering, though, if the product of this "undefined derivative" and zero is zero.
45. ### Continuity and Differentiability of Infinite Series

Homework Statement I came across a problem where f: (-π/2, π/2)→ℝ where f(x) = \sum\limits_{n=1}^\infty\frac{(sin(x))^n}{\sqrt(n)} The problem had three parts. The first was to prove the series was convergent ∀ x ∈ (-π/2, π/2) The second was to prove that the function f(x) was continuous...
46. ### Proving Twice Differentiability at a Point for a Function of Two Variables

Homework Statement Given f(x,y) = x\cdot 3^{x+y^2} . Prove that f is differentiable twice at the point P(1,0). Homework Equations D\subset\mathbb{R}^2, f\colon D\to\mathbb{R}, P\in \mathring{D}(interior point) - then f is differentiable n+1 times at P\Leftrightarrow \exists\varepsilon >...
47. ### How to Prove Differentiability in R2 Using the Derivative of a Function?

Let U={(x,y) in R2:x2+y2<4}, and let f(x,y)=√.(4−x2−y2) Prove that f is differentiable, and find its derivative. I do know how to prove it is differentiable at a specific point in R2, but I could not generalize it to prove it differentiable on R2. Any hint?
48. ### Differentiability implies continuous derivative?

We know differentiability implies continuity, and in 2 independent variables cases both partial derivatives fx and fy must be continuous functions in order for the primary function f(x,y) to be defined as differentiable. However in the case of 1 independent variable, is it possible for a...
49. ### Differentiability in an open and closed intervals

Is there an f(x) which is differentiable n times in a closed interval and (n+1) times in an open interval? I think I saw this in a paper related to Taylor's theorem (could be something else though). It didn't make sense to me, how can something be differentiable more in an interval that contains...
50. ### Differentiability of a function on a manifold

I am currently working through Nakahara's book, "Geometry, Topology and Physics", and have reached the stage at looking at calculus on manifolds. In the book he states that "The differentiability of a function f:M\rightarrow N is independent of the coordinate chart that we use". He shows this is...