My question relates to the properties of the Fourier series of a function, $f: \mathbb{R} \to \mathbb{R}$. I know from an elementary course in differential equations (for engineers) that, for all practical purposes, if $f$ satisfies the Dirichlet conditions, then the Fourier series of f, $\mathcal{S}(f)$, is equal to $f$ everywhere, except at jump discontinuities, where it equals the "average value of $f$." That's fine.
This leads me to wonder: supposing $f$ has finitely many jump discontinuities, is $\mathcal{S}(f)$ a continuous function? (Can we even talk about the function $\mathcal{S}(f)$ independently of $f$? It may be that my question stems from my own naivete.)
I asked my instructor and he said that, no, $\mathcal{S}(f)$ can't possibly be continuous since we know that, being equal to $f$ except at the jumps, that it has jumps discontinuities itself. I thought that sufficed.
However, $\mathcal{S}(f)$ is the sum of continuous functions (in general, sines, cosines, and a constant). Shouldn't it therefore be itself a continuous function? Granted, it is an infinite sum of continuous functions so it may be that I am unfamiliar with theory concerning the continuity properties of infinite sums of continuous functions.
Best Answer
Suppose we completely get rid of the assumption that $f$ has finitely many discontinuities and instead ask if the Fourier series of a continuous function is necessarily continuous. One can prove (using the techniques of functional analysis) that for pretty much all continuous functions $f$ on $[0,2\pi]$, the Fourier series of $f$ diverges at pretty much all of the points in $[0,2\pi]$ (both in the Baire category sense). A proof can be found in Rudin Real and Complex Analysis Thm 5.12.
A sufficient condition for pointwise convergence of the Fourier series to the function is that the function be Holder continuous. This is the reason why classical solutions to differential equations have the nice property that you can work with the Fourier series: typically such a solution will have a continuous first derivative, which implies Holder continuity of order 1. If you ever get to the point where you are working with non-classical solutions (which come up a lot--think about solving for a Dirac delta for example) then you typically cannot just work with the Fourier series pointwise.