Consider the continuous and non negative function $c : \mathbb R \to [0,1]$ defined by $$ c(x) = \begin{cases} \cos \frac{\pi x}{2} &\text{for } x \in [-1,1]\\ 0 &\text{otherwise} \end{cases}$$ Let also $(r_n)$ be an enumeration of the rational points of the interval $[0,1]$ and $(k_n)$ be a strictly increasing sequence of integers.

Based on those elements, one can build the sequence of functions $g_n : [0,1] \to [0,1]$ defined by $g_n(x) = c(k_n(x-r_n))$. Is it possible to have $$\lim\limits_{n \to \infty} g_n(y) = 0$$ for some $y \in [0,1]$?

The origin of the question is the construction of a sequence of continuous functions $g_n$ defined on $ [0, 1]$ such that $0 \le g_n \le 1$ and $$ \lim\limits_{n \to \infty} \int_0^1 g_n(x) \ dx = 0,$$ but such that the sequence $(g_n(x))$ converges for no $x \in [0,1]$.

This is question I raised at Mathematics.