Interactive Real Analysis - part of MathCS.org

For the given function we have || f_n(x) || = 1/n². Since 1/n² is a p-series with p = 2, so it converges. Therefore f_n(x) converges uniformly and absolutely, by Weierstrass' theorem. But since the individual functions f_n 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 f_n(x) is either 1 if x < 0, or 0 if x = 0, or ²/6 if x > 0.