One can find a limit of smooth functions which converge to the absolute value function which is not differentiable at $0$, as shown by the following image:
As we can see as the limit goes to infinity the functions start to become rounder and rounder and so the minima converge to a cusp.
This leads to an even more interesting question must the limit of continuous functions be continuous, it turns out that if these continuous functions converge *uniformly* then the answer is yes, but in general it is NO, as shown by the following example:
This sequence of functions is $f_{n}=\arctan(nx)$, this ends up converging to a function with a jump discontinuity!
The lesson to be dervived from these examples is that pointwise convergence is *extremely* badly behaved, and there is no reason to expect that the limit satisfies something all the functions of the sequence satisfy.
It should be noted that mathematicians had a hard time grasping this beofre more rigourous defintions were given to the notions described here.
The provided graphs were really helpful in visualization.
This is really insightful!
Great blog! It is great that you provide links to uniform convergence and point-wise convergence. Would be great if you could define these notions more formally and discuss the implications of uniform convergence in a separate blog