Sum of a infinite number of continuous functions on a set may not be continuous.

Question

I was asked to give an example of series of continuous functions whose limit is discontinuous .

I gave the following example: $f_n(x) = x^n - x^{(n-1)}$ . I thought any sequence of continuous functions which is convergent to a discontinuous function can be made by that way a series of continuous function whose limit is discontinuous. Am I correct?

Can anyone give me suggestion on this?

Answer 1

Yes! But here's the standard one:

Consider $$f_n(x)=\frac{x^2}{(1+x^2)^n}$$ where $x \in \Bbb{R}$ and $n=0,1,2,...$

Then $$f(x)=\sum_{n=0}^\infty f_n(x)= \sum_{n=0}^\infty\frac{x^2}{(1+x^2)^n}= \begin{cases} 0&x= 0\\ 1+x^2 &x\neq0\\ \end{cases} $$