currently taking Measure and Integration course, which seems to have a different definition of f'.
traditionally,
$$f'(x)=\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}$$
but in folland's book, it seems to be defined as
$$f'(x)=\lim_{r\to 0} \frac{f(x+r)-f(x-r)}{m(B(x,r))}$$
i was just wondering if these 2 definitions are really the same thing. thanks in advance
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$\begingroup$I'm assuming that $m(B(x,r))$ means $r$, because I can't make the question sensible any other way.
If the first limit exists then so does the second and they are equal. However, the second limit can exist while the first one doesn't.
For the first claim, note that $$\frac{f(x+r/2) - f(x-r/2)}{r} = \frac{1}{2} \left( \frac{f(x+r/2) - f(x)}{r/2} + \frac{f(x) - f(x-r/2)}{r/2} \right)$$ and use the standard result that, if $\lim_{t \to 0} g(t)$ and $\lim_{t \to 0} h(t)$ exist, then $\lim_{t \to 0} g(t) + h(t)$ exists and is the sum of the previous limits.
For the second claim, let $f(x) = |x|$. Then the derivative, defined in the usual sense, does not exist, but Folland's limit is $0$.
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