I'm confused about something I SHOULD know the answer to: Can a non-continuous function be differentiable? According to Desmos, it can. This is intuitive to me, as the y-value of a function does not affect the slope, so it doesn't seem like a vertical jump should affect the derivative.
However, I was also under the impression that, because the definition of derivative involves a nonzero $dx$, it is impossible to take the derivative at any endpoint. I suppose the one-sided limit of $\frac{f(x+h)-f(x)}{h}$ would exist, but the two-sided limit would not.
Also, a question from an old AP exam gives a graph of a derivative (no additional info) and the answer key makes the assumption that the original function is continuous. They seem to be implying that, since the derivative exists, the original function must be continuous.
To confuse me even more, I stumbled across separate terms differentiable and continuously differentiable.
Can someone help clarify? Thank you!
Best Answer
Yes: Differentiability (at a point) implies continuity at the point.
Differentiability on an interval implies continuity on an interval.
The derivative of a differentiable function is not necessarily a continuous function itself, however, as
$$ f(x) = \begin{cases}x^2 \sin(\frac1x) & x \ne 0 \\ 0 & x = 0 \end{cases} $$ shows (after you do a good deal of work).