I'm wondering if the following is a correct example of a continuous function which is differentiable nowhere:
Define $f_{n}(x)$=$sin(\pi t4^{n})$.
Then: 1) $\sum 2^{-n} f_{n}(t) $ exists, by Weierstrass's M test.
2) $h_{m}=2^{-m}$ is a sequence of real numbers which converges to 0.
3) $f_{n}(t+h_{m}) - f_{n}(t) =0 \forall n>\frac{m}{2}$
4) So, $\frac{f(t+h_{m}) - f(t)}{h_{m}}= \sum \limits_{n=1}^{floor(\frac{m}{2}))} \frac{f_{n}(x+h_{m}) - f_{n}(x)}{h_{m}} 2^{-n} < \sum 2^{m+1-n} =2^{m+2}(1-2^{-n+1})$ (by difference of sines of real arguments is less than 2) , and hence the limit of the above difference quotient as m goes to infinity (which guarantees h goes to 0) does not exist, for any x in Reals.
Thank You,
