Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

How should one test for differentiability of a function that is piecewise defined? How is the differentiability related to one sided derivative in this case? Also, in the limits $\lim_{h\to 0^+}$ or $\lim_{h\to 0^-}$ what do the positive and negative signs beside the zero indicate graphically?

2019年10月24日 · Determine differentiability for a piecewise function. Ask Question Asked 5 years, 4 months ago. Modified 5 ...

Determine continuity and differentiability. Ask Question Asked 11 years, 3 months ago. Modified 11 years ...

If the function is not one covered by previous results, then we have to discover new means for testing differentiability. This happens all the time, for example in beginning calculus. You differentiate polynomials, cover the sum, product, quotient, chain rules etc.

2017年6月2日 · I want to determine the differentiability of the following function at the origin $(0,0)$: \begin{equation} f(x,y)=\begin{cases} \frac{x^3y}{x^4+y^2} & x \neq 0, y\neq 0 \\ 0, & x=y=0\end{cases} \end{equation} I proved that the partial derivatives of the function, as well as its directional derivatives, at $(0,0)$ all equal zero using the limit ...

2020年6月22日 · $\begingroup$ For differentiability always use the definition : left and right hand derivatives should exist and be equal. $\endgroup$ – Paramanand Singh ♦ Commented Jun 23, 2020 at 7:53

2024年5月28日 · The "correct" way to determine differentiability is to use the definition of the derivative as a limit. However, sometimes we have a function with a piecewise definition. Say for example (just an example, the question itself is more general)

2014年9月19日 · Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

This will imply differentiability (although it is not a necessary condition). If you think a function is not differentiable in a given point, then you might be able to obtain a contradiction with the definition of differentiability by approaching the given point from various directions.