8.1. Pointwise Convergence
Example 8.1.8 (b): Pointwise Convergence does not preserve Differentiability
We first need a function that is continuous but not differentiable. Then we will try to find a sequence of differentiable functions that converge to it.
Our prime example of a nondifferentiable, continuous function is, of course, the absolute value function f(x) = x. To approximate it by differentiable functions we need to "smooth out" the "kink" at x = 0, for example with a piece of a parabola. In other words, we need a function defined in multiple pieces:
 to the left of x=0 the function should look similar to the negative part of the absolute value, i.e. f(x) = x
 to the right of x=0 the function should be close to the positive part of the absolute value, i.e. f(x) = x
 in a small neighborhood of x=0 the function should look like a parabola f(x) = a x^{2} for some positive constant a
 of course the function should depend on a parameter n in
such a way that:
 the function is differentiable
 the function is continuous
 the function is 'close' to x
Let's decide that the 'small neighborhood' mentioned in part 3 is the interval (1/n, 1/n), inside which the function is a parabola. To ensure its derivative matches up with the derivatives outside that interval, we need f'(x) = 2a x = 1 when x 1/n. Thus, a = n/2. Finally we need to patch our function to make it continuous at x 1/n. Our finished product is:


You should verify that this function has all of the desired properties:
 f_{n}(x) is continuous at 1/n and in fact in [1, 1]
 f_{n}(x) is differentiable at 1/n and in fact in [1, 1]
 f_{n}(x) converges to f(x)=x