Hi all. Doing a problem led me to derive this little lemma, but I'd like someone to check it for me if you would be so kind.

Theorem: Let f be an integrable function and define F(x) = \int_{0}^{x} f(t)\,dt. Suppose that, for some c in the domain of f we have \lim\limits_{x \rightarrow c^{-}} f(x) \neq \lim\limits_{x\rightarrow c^{+}} f(x), where the limits are presumed to exist. Then F(x) is not differentiable at c.

Proof: Assume without loss of generality that \lim\limits_{x \rightarrow c^{-}} f(x) = \ell_{1} < \ell_{2} \lim\limits_{x\rightarrow c^{+}} f(x). Then there is some \delta>0 such that 0 < |x-a| < \delta implies:
  1. f(x) < \ell_{1} + \frac{\ell_{2}-\ell_{1}}{4} = m for x<c
  2. f(x) < \ell_{2} - \frac{\ell_{2}-\ell_{1}}{4} = M for x>c

Clearly M>m. Then for any 0<h<\delta we have

 \frac{F(x+h) - F(x)}{h} = \frac{1}{h} \int_{x}^{x+h} f(t)\,dt > \frac{Mh}{h} = M
and for any -\delta <  h < 0 we have

 \frac{F(x+h) - F(x)}{h} = -\frac{1}{h} \int_{x+h}^{x} f(t)\,dt < -\frac{1}{h} \cdot -mh = m
so that \lim\limits_{h\rightarrow 0} \frac{F(x+h)-F(x)}{h} cannot exist. QED.

Any help is much appreciated!