Consider the series $f(x)=\sum_{n=1}^{\infty}e^{-nx^{2}}\sin(nx)$:
a) Prove that this series converges uniformly on $[a,\infty)$, for each $a>0$
b) Does the series converge uniformly on $[0,\infty)?$
We can show uniform convergence on the interval $[a,\infty)$ with Weierstrass M-Test for part (a). For part (b), I suspect that there's a discontinuity at $x=0$ but I don't know how to justify it mathematically.