8.2. Uniform Convergence
Example 8.2.6 (b): Uniform Convergence does not imply Differentiability
Find a sequence of differentiable functions that converges uniformly to a
continuous limit function but the limit function is not differentiable
Before we found a sequence of differentiable functions that
converged pointwise to the continuous, nondifferentiable function f(x) = x.
Recall:


That same sequence also converges uniformly, which we will see by looking at `  f_{n}  f_{D}. We will find the sup in three steps:
 If 1 x 1/n:
  f_{n}(x)  f(x) = x  ^{1}/_{2n} + x = ^{1}/_{2n}
 If 1/n < x < 1/n:
 f_{n}(x)  f(x) ^{n}/_{2} x^{2} + x ^{n}/_{2} ^{1}/_{n2} + ^{1}/_{n} = ^{3}/_{2n}
 If 1/n x 1:
  f_{n}(x)  f(x) = x  ^{1}/_{2n}  x = ^{1}/_{2n}
Thus,  f_{n}  f_{D} < ^{3}/_{2n} which implies that f_{n} converges uniformly to f. Note that all f_{n} are continuous so that the limit function must also be continuous (which it is). But clearly f(x) = x is not differentiable at x=0.