What does differentiability imply about local linear approximation?

Near an interior point $a$, $f(a+h)=f(a)+f'(a)h+r(h)$, where $r(h)/|h|\to 0$ as $h\to 0$.

What is a one-sided derivative?

A derivative defined by a difference-quotient limit using inputs that approach the point from one direction.

By a Jacobian matrix acting as a linear map on small input changes.

The first-order linear map must be multiplication by a complex number, with the quotient limit agreeing in every direction.

The function's coordinate representation must be differentiable in suitable local charts.

Real analysisLesson 5 of 5 in the course Continuity and differentiability

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This lesson introduces the derivative as the defining test for differentiability and connects it to the local shape of a graph.

For a real function $f$ defined on an open set $U\subseteq\mathbb{R}$ , the function is differentiable at $a\in U$ when the limit

$f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$

exists as a finite real number. Equivalently, one may let $x$ approach $a$ and use $\frac{f(x)-f(a)}{x-a}$ .

What does differentiability imply about local linear approximation?: Near an interior point $a$ , $f(a+h)=f(a)+f'(a)h+r(h)$ , where $r(h)/|h|\to 0$ as $h\to 0$ .
What is a one-sided derivative?: A derivative defined by a difference-quotient limit using inputs that approach the point from one direction.
How is the derivative of a differentiable multivariable function represented?: By a Jacobian matrix acting as a linear map on small input changes.
What special requirement does complex differentiability impose on the linear approximation?: The first-order linear map must be multiplication by a complex number, with the quotient limit agreeing in every direction.
How is differentiability defined for functions on manifolds?: The function's coordinate representation must be differentiable in suitable local charts.

For a function $f:\mathbb{R}^m\to\mathbb{R}^n$ , what condition characterizes differentiability at $\mathbf{x}_0$ ?Answer: There is a linear map $M$ such that the norm of the approximation error divided by $\|\mathbf{h}\|$ tends to zero.
A function has all its partial derivatives at a point. What can be concluded from that fact alone?Answer: Differentiability at that point is not guaranteed.
For a complex function to have a complex derivative at $a$ , what must happen as $h\to 0$ ?Answer: The difference quotient must approach the same limit for every direction of approach in $\mathbb{C}$ .
When is a complex function holomorphic at a point?Answer: When it is complex-differentiable throughout a neighborhood of that point.
If a real function is differentiable at an interior point, what must also hold there?Answer: The function is continuous at that point.

Based on the Wikipedia article Differentiable function, licensed CC BY-SA 4.0. Adapted into a summary, flashcards and quizzes.