Let $k>0$ and $g_t$ be a gaussian-type function with center $t$ defined by $g_t(x):=e^{-k(x-t)^2}$ for all $x\in \mathbb R$. I intuitively observe that the map $g: t\in \mathbb R \longrightarrow g_t\in L_\infty(\mathbb R)$ is continuous, i.e. $$f(t):=\max_{x\in \mathbb R} |e^{-k(x-t)^2} -e^{-kx^2}|\rightarrow 0 \text{ if } t\rightarrow 0.$$
I think that we should find a good estimation to prove it. What I tried is to observe that $e^{-kx^2}\leq 1$, then $f(t)\leq \max_{x\in \mathbb R} |e^{2kxt-t^2} -1|$. But for each fixed $t\in \mathbb R$, the right-hand-side is infinity. So this estimation is useless. How do I do a better estimation?