Proving the Continuity of $e^x$ Can one say that $e^x$ is the sum of an infinite number of terms (Taylor expansion), every term being a continuous polynomial in itself, the sum of all the terms is continuous and so $e^x$ is continuous?
Thanks in advance
 A: Infinitely many continuous functions added together doesn't imply the result is also continuous. Take the Fourier series of a square wave function for example. All terms are continuous but the result is not.
$$f(x)=\frac{4}{\pi}\sum_{n=1,3,5,...}{\frac{1}{n}\sin(\frac{n\pi x}{L})}$$
A: Here's another approach. Since $\exp x:=\sum_{n\ge0}\frac{x^n}{n!}$ converges for all $x$ by the ratio test, and you can show $\exp x\exp h=\exp (x+h)$ with the binomial theorem, we need only show continuity at $0$, since if $|h|<\delta\implies|\exp h-1|<\frac{\epsilon}{\exp x}$ then $|h|<\delta\implies|\exp(x+h)-\exp x|<\epsilon$. By the triangle inequality,$$|\exp h-1|=\left|\sum_{n\ge1}\frac{h^n}{n!}\right|\le\sum_{n\ge1}\frac{|h|^n}{n!}\le2\sum_{n\ge1}\left(\frac{|h|}{2}\right)^n=\frac{2|h|}{2-|h|}\le 2|h|$$for small $h$.
A: $$e^x=\sum_{n=0}^{+\infty}\frac{x^n}{n!}$$
the Radius of convergence is
$$R=\lim_{n\to+\infty}\frac{(n+1)!}{n!}=+\infty$$
thus the sum of the power series is infinitely differentiable, and of course continuous at
$$(-R,R)=(-\infty,+\infty).$$
but, for example
$\sum_{n=0}^{\infty}x^n$ is not continuous at $x=1$.
A: Another prove different from the prove of Hamam can be to prove that $e^x$ is a derivable function, and so it must be also continuos:
$\lim_{x\to x_0}\frac{e^x-e^{x_0}}{x-x_0}=$
$\lim_{x\to x_0}e^{x_0}\frac{e^{x-x_0}-1}{x-x_0}=e^{x_0}\lim_{y\to 0}\frac{e^y-1}{y}=e^{x_0}$
so $e^x$ is derivable in each point and it must be continuos in each point.
Your proof is not correct in general and a counter example is the function  posted by Sep. In general you can say that a series of continuos function is continuos at most in a compact set, because in general the space of continuos functions on a compact set is closed in the space of bounded functions with respect $\infty -$ norm. 
A: It is true that


*

*The sum of a finite number of continuous functions is a continuous function.

*The product of a finite number of continuous functions is a continuous function.


To prove these, assume that $f(x)$ and $g(x)$ are continuous functions at a point $a$. Then,
$$\lim_{x\to a}f(x)=f(a)$$
$$\lim_{x\to a}g(x)=g(a)$$
So, 
$$\lim_{x\to a}f(x)\cdot g(x)=\lim_{x\to a}f(x)\cdot \lim_{x\to a}g(x)=f(a)\cdot g(a)$$
and
$$\lim_{x\to a}\Big(f(x)+ g(x)\Big)=\lim_{x\to a}f(x)+ \lim_{x\to a}g(x)=f(a)+ g(a)$$
But, the sum or product of infinitely many continuous functions isn't necessarily a continuous function.
