Finding the limit of a weird sequence function? Do you know any convergent sequence of continuous functions, that his limit function is a discontinuous  function in infinitely many points of its domain (?) 

Let $$f_n(x) = \begin{cases} nx & \hat{x}\in[0,\frac{1}{n}) \\ 1 & \hat{x}\in[\frac{1}{n},\frac{n-1}{n}) \\ n-nx & \hat{x}\in(\frac{n-1}{n},1)\end{cases},$$ where $\hat{x}$ is the fractional part of $x$. It is not to hard to see that $f_n$ is continuous for every $n\in\mathbb{N}$, but $\lim_{n\to\infty} f_n = f_\infty$, where $f_\infty(x) = \begin{cases} 1 & x\notin\mathbb{Z} \\ 0 & x\in\mathbb{Z}\end{cases}$, which is clearly not continuous at any integer.
