Consider the function $f \in C_{st}$ that satisfies that $$ f(x) = 6x+2 $$ Then I have to argue for or against that the Fourier series for $f$ converges pointwise or uniformly on $\mathbb{R}$.
I have said that as $f(x)$ is a well-known continuous function we can calculate $$ \lim_{x \rightarrow \pi^-} 6x+2 = 6\pi + 1 $$ and $$ \lim_{x \rightarrow -\pi^+} 6x+2 = -6\pi + 1 $$ Thus $f$ is continuous at $\pm \pi$ and $f(\pi) = 6\pi + 1$ and $f(-\pi) = -6\pi +1$. But doesn't this mean that $f$ is not $2\pi$ periodic? It just does not make sense to me as one of the criteria for being a function in $C_{st}$ is indeed that the function $f$ is $2\pi$ periodic.
Thus as $f$ is not $2\pi $-periodic, $f$ does not converge either pointwise or uniformly on $\mathbb{R}$. Is this correct? But how can $f$ be in $C_{st}$ then?