A sequence of continuous functions which is pointwise convergent to zero and not uniformly convergent on any interval. The exercise is to construct a sequence of continuous functions $f_n:\mathbb{R}\rightarrow \mathbb{R}, n\in \mathbb{N}$ , which is pointwise convergent to $f(x)=0 , x\in \mathbb{R}$ and not uniformly convergent on any interval $(a,b)$.
I posted this question once and I got the answer, but I had forgotten to mention that $f_n$ must be continuous. The example of sequence of discontinuous functions looks like this:

Let $\{r_n\}_{n\geq 1}$ be an enumeration of the rationals, and define $f_n(x)$ as follows:
  $$
f_n(x)=\begin{cases}
1,& x\in \{r_k\}_{k\geq n}\\
0,& else
\end{cases}
$$

The person, who posted this, made the remark: 

If you want to make the $\{f_n\}$ continuous, you can modify this example by using bump functions supported on $(r_n-2^{-n},r_n+2^{-n})$ in place of the spikes to $1$ at $r_n$.

I'm not realy sure what he meant. I was thinking of defining $g_k$ as a broken line or maybe more smooth function going through $(r_k-2^{-k},0), (r_k,1), (r_k+2^{-k},0)$ and zero on $\mathbb{R}\setminus [r_k-2^{-k},r_k+2^{-k}]$ and then defining $f_n(x)=\sup\{g_k(x):k\ge n\}$.  But I don't know whether it converges to zero and whether $f_n$ are even continuous.
On the other hand I have found this topic
A sequence of continuous functions on $[0,1]$ which converge pointwise a.e. but does not converge uniformly on any interval
and there is only an example of a sequence converging to zero almost everywhere. That's why I begin to doubt that there exists a sequence converging to zero everywhere (that would mean there is a mistake in my book because the exercise says "construct such a sequence").

According to the hint in "Problems in Mathematical Analysis", Biler and Witkowski, problem 4.112.
Define $g_n$ as the function which is $0$ on the intervals $(-\infty, 0]$ and $[2/n,\infty)$, $1$ at $1/n$, and linear otherwise.  Then $g_n$ converges pointwise to $0$ but is not uniform in any interval containing $0$.
Now define $$f_n(x) = \sum_{k=0}^{\infty} 2^{-k} g_n(x - r_k).$$
Then the $f_n$ are continuous, converge pointwise to $0$, but do not converge uniformly on any open interval. 
