How do I determine if this function is uniformly continuous at this interval?

Question

How do I determine if

$$f(x)=\pi^{x^2}$$

is uniformly continuous at $$(-\infty,\infty)$$

If the functions derivative had a max limit, then I could easly prove, but I does not have. So what should I do?

Answer 1

Since $\lim_{x\to\infty}f'(x)=\infty$, the function $f$ is not uniformly continuous. In fact, if $\delta>0$, take $x_0\in\mathbb R$ such that $x\geqslant x_0\implies f'(x)>\frac1\delta$. Then, by the mean value theorem,$$f(x_0+\delta)-f(x_0)\geqslant\delta\times\frac1\delta=1.$$