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...