My professor said in class:
The derivative of f at x=a is f'(a) = [f(a+h) - f(a) ] /h (and said that this Limit does not need to exist for the derivative to exist). (Then he said) If f'(a) exists, then f is differentiable at x=a
Can someone please give me an example of a case where the limit does not need to exist at x=a for the function to have a derivative at x=a ?

If f'(a) exists, does that mean f is continuous on x=a? ( I think yes) But this is not the other way around?

2. ## Re: Clarification about Differentiation

I am NOT going to assume that you are quoting your professor correctly since he/she is not here to define him/herself! No, [f(x+ h)- f(x)]/h is NOT the derivative- it is the "difference quotient" and the derivative is the limit of that as h goes to 0. That limit must exist in order that the function be "differentiable" there.

Now, one of the basic properties of limits is that if both $\lim_{x\to a} f(x)$ and $\lim_{x\to a} g(x)$ exist, then $\lim_{x\to a}\frac{f(x)}{g(x)}$ exists. However, for the difference quotient, since the denominator is h, the denominator will always go to 0- but the limit may exist anyway. Perhaps that is what your professor was trying to tell you.

3. ## Re: Clarification about Differentiation

Originally Posted by sakonpure6
My professor said in class:

Can someone please give me an example of a case where the limit does not need to exist at x=a for the function to have a derivative at x=a ?

If f'(a) exists, does that mean f is continuous on x=a? ( I think yes) But this is not the other way around?

It depends on what you mean by derivative, in the distributional sense the derivative of a step is a dirac delta, but here the derivative of distributions is not defined as a limit but by the identity:

$$\int_{-\infty}^{\infty}f'g =-\int_{-\infty}^{\infty}fg'$$

where $f$ is our distribution and $g$ a tempered distribution (infinitely differentiable and goes to zero (with all its derivatives) as $x$ goes to $\pm \infty$) with the usual definition of derivative for tempered distributions.

.

4. ## Re: Clarification about Differentiation

I think either you misheard or your teacher misspoke.

$\displaystyle \lim_{h \rightarrow 0}\dfrac{f(a + h) - f(a)}{h}\ may\ or\ may\ not\ exist,\ but\ \lim_{h \rightarrow 0}\dfrac{f(a + h) - f(a)}{h}\ exists \iff f(x)\ is\ differentiable\ at\ a.$

And differentiability does imply continuity, but continuity does not imply differentiability.