Example of a sequence of functions where the limit cannot be interchanged 
Give an example of a sequence of continuous functions $f_n$ on $[0,1]$ with $f_n$  converges pointwise to a continuous function $f$ such that the following relation does't hold:
$$\lim_{n \rightarrow \infty} \lim_{x \rightarrow 0} f_n(x)=\lim_{x \rightarrow 0}\lim_{n \rightarrow \infty} f_n(x)$$

I know such a convergence is not uniform. I already tried with this one: $f_n(x)= 2nx e^{-nx^2}$. Actually this one satisfies the given limit condition even though the convergence is not uniform! Any hint?

Since all $f_n$ are continuous we have $\lim_{x\to0} f_n(x) = f_n(0)$ and since $f = \lim_{n\to\infty} f_n$ is also continuous we have $\lim_{x\to 0} f(x) = f(0)$.
So your relation boils down to $\lim_{n\to\infty} f_n(0) = f(0)$, which is true because $f_n \to f$ pointwise. 
