I calculated the Fourier coefficients of the square wave.
$$ f(x) = \begin{cases} -1, & x\in [-\pi, 0) \\ 1, & x\in [0, \pi] \end{cases} $$
Then I came to the conclusion that the $N^{th}$ partial Fourier sum is given by
$$ S_{N,f}(x) = \frac{4}{\pi} \sum_{k \ \text{odd}, \ k > 0 }^{N} \frac{\sin(kx)}{k} $$
Now does $S_{N,f}(x)$ converge to $f(x)$ uniformly?
What are some usefull theorems I can use here? Or must I prove it by def. of uniformly convergence? I found a theorem called Weierstrass M test. But I dont think it is usefull here, since the only bound for $|\frac{\sin(kx)}{k}|$ that I can come up with is $\frac{1}{k}$ which is not helpful.
If a sequence of continuous functions converges uniformly, they converge to a continuous function (see the uniform convergence theorem). Since $\lim\limits_{n\to\infty}S_{n,f}$ is not continuous, the convergence cannot be uniform even if you redefine your function to be $0$ at $x=0$.