As the composition sum of differentiable functions, shouldn't it also be differentiable? I am aware that a square wave is non-continuous non-smooth and thus not differentiable, I am just having trouble finding the flaw in the above argument.
Edited inaccuracies that were pointed out!
The square wave is the sum of an infinite number of differentiable functions, and the sum of an infinite number of differentiable functions is not necessarily differentiable.
To see why this happens in this case, consider the partial sums for the Fourier series: $$ f_m(x) = \sum_{n = 1}^{m} \frac{4}{(2n-1)\pi} \sin ((2n-1)\pi x) $$ Since the partial sums are finite sums, we can take their derivatives, to see that $$ f'_m(x) = \sum_{n = 1}^{m} 4 \cos ((2n-1)\pi x). $$ In particular, $f'_m(0) = 4m$, and $\lim_{m \to \infty} f'_m(0) = \infty$.