Example 8.3.4 (c): Function Series Examples
For the given function we have || fn(x) || = 1/n2. Since 1/n2 is a p-series with p = 2, so it converges. Therefore fn(x) converges uniformly and absolutely, by Weierstrass' theorem. But since the individual functions fn are not continuous, the sum is not necessarily continuous.
In fact we know that 1/2n = 1 and 1/n2 = 2/6 (as we will see later), so fn(x) is either 1 if x < 0, or 0 if x = 0, or 2/6 if x > 0.