Uniform Convergence Proof in Spivak's Calculus

Question

I have done everything in this problem except "conclude that the series does not converge uniformly on $\mathbb{R}$". I know something happens at $x=0$ and that $$ \lim_{N \to \infty} f \bigg( \frac{1}{N}\bigg) \geq \frac{1}{4} $$ might be useful, but I am not sure where to go from here. How can this be shown?

Answer 1

Since $f(0)=0$, the limit shows that $f$ is not continuous. Actually the limit likely does not exist; all you are showing is that $f(1/n)\geq1/4$ for all $n$, not that the limit exist. Still, this is enough to show that $f$ is not continuous.

In any case, if the series were to converge uniformly on an interval containing zero, its limit would be continuous in that interval.