Whether in Real Analysis or by Open Set Def of Continuity in Topology, it is easy to show that the sum and product of a FINITE number of continuous functions are also continuous functions. That is, assuming that $f_1, …, f_m:\Bbb R\rightarrow\Bbb R$ are continuous, then $S:\Bbb R\rightarrow\Bbb R$ and $P:\Bbb R\rightarrow\Bbb R$, defined by $S(x) = f_1(x) + … + f_m(x)$ and $P(x) = f_1(x) \times … \times f_m(x)$, are continuous.
But many analytic functions that are continuous can be written in their expanded form (by Taylor Series), which are the sum and product of INFINITE functions.
My question is, EVEN if there is another way to show that some/all analytic functions are continuous (which I don't know that way), still we should prove from "the sum and product of infinite functions" way.
Would you please help me regarding the question? I think one of Topology or Analysis way of proof should be enough, because, as we can prove, the topological definition of continuity is equivalent to the $\epsilon – \delta$ definition for functions that map $\Bbb R$ to $\Bbb R$.
EDIT: Let me rephrase it: limit of sum of two functions exists if limit of each of the two functions exists. If sum of in finite number of functions is a function that has limit in some point, is it mean that we are allowed to say that for this type of function, sum of limit infinite number of functions exists since limit of sum of those infinite number of functions exists?
Best Answer
Consider the function defined by $$\begin{eqnarray}f(x) &= &\begin{cases} 1 &\mbox{if}\quad 0<x<\pi\\ 0 &\mbox{if}\quad x=0\\ -1 &\mbox{if}\quad -\pi<x<0 \end{cases}\\ f(x+2n\pi) &=& f(x) \quad \mbox{if}\quad n \in \mathbb N \end{eqnarray}.$$ This is an example of a "square wave." It is not continuous, but Fourier analysis gives an infinite sum of functions (all of the form $a \sin(k\pi)$) that add up to this function.
Regarding your edited question, on the other hand, the only way I know of to even define the sum of an infinite number of functions is that the sum is the limit function $f$ of the sequence of finite function sums $f_n$ as $n$ goes to infinity, if such a function $f$ exists. If we know that $f$ has a limit at a certain point, that is, $\lim_{x\to\infty} f(x)$ exists for a certain value of $x$, then $f$ is continuous at $x$, and so is the infinite sum of functions (since that sum is defined as $f$).