# 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. ### A Vector calculus - Prove a function is not differentiable at (0,0)

##f\left(x\right)=\begin{cases}\sqrt{\left|xy\right|}sin\left(\frac{1}{xy}\right)&xy\ne 0\\ 0&xy=0\end{cases}## I showed it partial derivatives exist at ##(0,0)##, also it is continuous as ##(0,0)## but now I have to show if it differentiable or not at ##(0,0)##. According to answers it is not...
2. ### B Difference between a continuously differentiable function and a wave

What is the difference between an absolutely continuously differentiable function and a wave? Are all absolutely continuously differentiable equations waves?
3. ### MHB Is f differentiable in (0,0) ?

Hey! :giggle: We consider the function\begin{align*}f:\mathbb{R}^2 &\rightarrow\mathbb{R} \\ (x,y)&\mapsto \begin{cases}\frac{x^3}{x^2+y^2} & \text{ if } (x,y)\neq(0,0) \\ 0 & \text{ if } (x,y)=(0,0) \end{cases}\end{align*} (a) Show that all directional derivatives of $f$ in $(0,0)$ exist...

10. ### A Softened potential well / potential step

Do any of you know of an article or book chapter that discusses the difference between a discontinuous potential well of length ##2L## ##V(x)=\left\{\begin{array}{cc}0, & |x-x_0 |<L\\V_0 & |x-x_0 |\geq L\end{array}\right.## and a differentiable one ##\displaystyle V(x) = V_0...
11. ### I An interesting point regarding critical points and extrema

Hi all, I have recently faced some problem about distances between two curves, and (re?)"discovered" an interesting point that I would like to share with you. In the following, we consider a function of two variables ##f(x,y)##, but it should be clear that the definitions and the result is...
12. ### Let function ƒ be Differentiable

What I've tried is: I have defined a function g(x)=f(x)-x^2/2. g Differentiable in the interval [0,1] As a difference of function in the interval. so -x≤g'(x)≤1-x for all x∈[0,1] than -1≤g'(x)≤0 or 0≤g'(x)≤1 . Then use the Intermediate value theorem . The problem is I am not given that f' is...
13. ### Multi-Choice Question: Differentiable function

be f Differentiable function In section [0,1] and f(0)=0, f(1)=1. so: a. f A monotonous function arises in section [0,1]. b. There is a point c∈[0,1] so that f'(c)=1. c. There is a point c∈(0,1) where f has Local max. I have to choose one correct answer.
14. ### I Differentiable manifolds over fields other than R, C

[Moderator's note: Spin-off from another thread.] You need the structure of a topological vector field K with 0 as a limit point of K-{0}. The TVF structure allows the addition and quotient expression to make sense; you need 0 as a limit point to define the limit as h-->0 and the topology to...
15. ### I How do charts on differentiable manifolds have derivatives without a metric?

I was reading about differentiable manifolds on wikipedia, and in the definition it never specifies that the differentiable manifold has a metric on it. I understand that you can set up limits of functions in topological spaces without a metric being defined, but my understanding of derivatives...
16. ### Prove the dipole potential is differentiable everywhere except at the surface

The dipole potential is given by: ##\displaystyle\psi=\int_{V'} \dfrac{\rho}{|\mathbf{r}-\mathbf{r'}|} dV' +\oint_{S'} \dfrac{\sigma}{|\mathbf{r}-\mathbf{r'}|} dS'## I need to prove that ##\psi## is differentiable at points except at boundary ##S'## (where it is discontinuous) I know if...
17. ### A What is a differentiable variety?

In mathematics, variety is a generalization of the surface idea. There are several types of varieties, according to the properties they possess. The most usual are the topological varieties and the differentiable varieties. but I still do not know what it is!
18. ### B Differentiable function - definition on a manifold

Hi, a basic question related to differential manifold definition. Leveraging on the atlas's charts ##\left\{(U_i,\varphi_i)\right\} ## we actually define on ##M## the notion of differentiable function. Now take a specific chart ##\left(U,\varphi \right)## and consider a function ##f## defined...
19. ### Show that a continuously differentiable function is not 1-1

Homework Statement "Let ##f:ℝ^2\rightarrow ℝ## be a continuously differentiable function. Show that ##f## is not one-to-one." Homework Equations A function ##f:ℝ^n\rightarrow ℝ^m## is continuously differentiable if all the partial derivatives of all the components of ##f## exist and are...

36. ### Find A and B so that F(x) is a Differentiable Function

Homework Statement Find the values of a and b that make f a differentiable function. Note: F(x) is a piecewise function f(x): Ax^2 - Bx, X ≤ 1 Alnx + B, X > 1 Homework Equations The Attempt at a Solution Made the two equations equal each other. Ax^2 - Bx = Alnx + B Inserting x=1 gives, A -...
37. ### Expressing defined integral as composition of differentiable functions

Homework Statement Let ##f(t)=\int_{t}^{t^2} \frac{1}{s+\sin{s}}ds,t>1.##Express ##f## as a composition of two differentiable functions ##g:ℝ→ℝ^2## and ##h:ℝ^2→ℝ##. In addition, find the derivative of ##f## (using the composition). Homework Equations The Attempt at a Solution Honestly, I...
38. ### Prove a statement regarding differentiable mv-function

Homework Statement Let ##f##: ##G\subset\mathbb{R}^2\rightarrow\mathbb{R}## be differentiable at ##(x_0,y_0)\in{G}## and ## \lim_{(x, y) \to (x_0, y_0)} \frac{f(x,y) -a -b(x-x_0) -c(y-y_0)}{\sqrt{(x-x_0)^{2} + (y-y_0)^{2}}} = 0.## The task is to prove that then ##a=f(x_0,y_0),b=f_x(x_0,y_0)##...
39. ### MHB Differentiable Function

Hello all, I am not sure how to approach this question: Let f(x) be a continuous and differentiable function of order 2. Let f''(x) >0 for all values of x. The tangent line to the function at x=1 is y=-x+1. Show that f(1.1)>-0.1. Thanks!
40. M

### B Continuous and differentiable functions

"If a function can be differentiated, it is a continuous function" By contraposition: "If a function is not continuous, it cannot be differentiated" Here comes the question: Is the following statement true? "If a function is not right(left) continuous in a certain point a, then the function...
41. ### MHB Differentiable function

Hello! (Wave) Suppose that we want to check if $f(x,y)=\frac{x}{y}+\frac{y}{x}$ is differentiable at each point of its domain and if it is $C^1$. The domain is $D=\{ (x,y) \in \mathbb{R}^2: x \neq 0 \text{ and } y \neq 0\}$. The partial derivatives ...
42. ### B Continuous but Not Differentiable

Suppose a certain function in continuous at c and (c. f(c)) exists, then which of the two could be false: \displaystyle \lim_{x \rightarrow c^-} {f(x)} = \lim_{x \rightarrow c^+} {f(x)}, and \displaystyle f'(c)? I feel like both could be false, because if the formal derivative at a point...
43. ### 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?
44. ### Is the function differentiable at x = p?

Hello mates, is the function ## f(x) = \frac{(2^x - 1)}{x} ## differentiable at x = 0? For it to be differentiable it has to be continuous? From the graph f(0) is undefined although limit exists. I have read that at points like a corner, gap and vertical tangents it is not differentiable. So...
45. ### Real-Life Signals: Are They Infinitely Continuous & Differentiable?

Are all real life signals infinitely continuous and differentiable? I'm thinking yes because a finite discontinuity in one of the derivatives would imply infinite to take place in the next higher-order derivative. And infinite means infinite energy.
46. ### Differentiable Manifolds

Hello every one Can one say , that A globle coordinate chart is a cartesian coordinate And a local coordinate chart is any kind of curvilinear coordinate ? Thanks
47. ### Insights A Continuous, Nowhere Differentiable Function: Part 2 - comments

jbunniii submitted a new PF Insights post A Continuous, Nowhere Differentiable Function: Part 2 Continue reading the Original PF Insights Post.
48. ### Insights A continuous, nowhere differentiable function: Part 1

jbunniii submitted a new PF Insights post A Continuous, Nowhere Differentiable Function: Part 1 Continue reading the Original PF Insights Post.
49. ### MHB Differentiable Curves

Hey! :o How could we prove the following rule for differentiable curves in $\mathbb{R}^3$ ?? (Wondering) \frac{d}{dt}[\overrightarrow{\sigma}(t)\times \overrightarrow{\rho}(t)]=\frac{d\overrightarrow{\sigma}}{dt}\times \overrightarrow{\rho}(t)+\overrightarrow{\sigma}(t)\times...
50. ### MHB PDE or differentiable manifolds?

Hello! :o I am doing my master in the field Mathematics in Computer Science. I am having a dilemma whether to take the subject Partial differential equations- Theory of weak solutions or the subject differentiable manifolds. Could you give me some information about these subjects...