I asked a similar question here but didn't get a full explanation, so I would like to ask a simpler related question.
The sequence of continuous functions \begin{equation*} f_n(x)= \begin{cases} x^n,\quad x\in[0,1)\\ 1,\quad x=1 \end{cases} \end{equation*} converges pointwise to \begin{equation*} f(x)= \begin{cases} 0,\quad x\in[0,1)\\ 1,\quad x=1 \end{cases}, \end{equation*} which is not continuous.
In particular, we know that uniform convergence is needed to guarantee that the limit is continuous. But this is a sequence of continuous functions on $[0,1]$, shouldn't its limit be continuous?
As user 5xum says in their answer, no: the limit of a sequence of continuous functions in not required to be continuous. In this case you can get a feel for what's happening geometrically by drawing the graph of $$f_n = x^n$$ on the interval $[0,1]$ for several values of $n$; say $n=1,10,100$ and $1000$. You'll see that the curve flattens towards the start of the interval, getting closer to the $x$-axis, and steepens towards the end of the interval, becoming closer and closer to vertical. It's as though in the limit the curve finally reaches $90^\circ$ and snaps, so that the last point separates from an otherwise flat curve.
Generalising this a little leads to the notion of Baire classes of functions. Continuous functions form Baire class 0, and the limits of continuous functions form Baire class 1, and we can build them up into a hierarchy. (Which turns out not to contain all possible functions either, so there exists non-Baire-class functions though they're a little tricky to conceptualise.) An interesting fact is that the derivative of a differentiable function is a Baire class one function.