##f_n(x)=x^n## are continuous functions.
$$
\displaystyle{f(x):=\lim_{n \to \infty}f_n(x)=\lim_{n \to \infty}x^n}=\begin{cases}0&\text{ if }0\leq x< 1\\1&\text{ if }x=1\end{cases}
$$
##f(x)## is not continuous since it jumps from ##0## for all values left of ##x=1## to ##1## for ##x=1.## This is a discontinuity.
Another example:
If we walk the unit square from bottom left to top right then we have to walk 2 units.
This does not change if we walk half a step to the right, then half a step to the top, another half of a step to the right, and finally another half to the top. That makes 2 units in total.
This does not change if we walk a quarter of a step to the right, then a quarter of a step to the top, another quarter, and so on until we end up at the top right corner. Still 2 units in total.
This does not change if we walk an eighth of a step to the right, another eighth of a step to the top etc.
No matter how small our steps will be. As long as we walk only to the right and to the top, we will have to walk 2 units in total. Nevertheless, we constantly approach the diagonal of the unit square which is of length ##\sqrt{2}\neq 2.## Hence, no matter how close we get pointwise to the diagonal, our path is of length 2 whereas we get pointwise closer to the diagonal of length ##\sqrt{2}.##