I am trying to prove or disprove $f(x) = e^{x^{2}}$ and $g(x) = e^{-x^{2}}$ is uniformly continuous for $x \in (- \infty, \infty)$.
My book says a function is uniformly continuous on an interval $I$ if for every number $\epsilon \gt 0$ there exists a $\delta = \delta(\epsilon)$ such that $\left\{ x, x_0 \in I \text{ and } |x – x_0| \lt \delta \right\} \Rightarrow |f(x) – f(x_0)| \lt \epsilon$. So apparently all I need to do in order to prove uniform continuity is to find a $\delta$ that is independent of $x_0$. I have seen a lot of similiar examples but the choice of $\delta$ always seem rather arbitrary to me.
I have some stuff that I am allowed to refer to as well:
$\large{\text{Theorem 1}}:$ If $f$ is continuous on the closed, finite interval $\left[a, b\right]$, then $f$ is uniformly continuous on that interval.
$\large{\text{Satz 1}}:$ Assume $f$ is differentiable and that the derivative is bounded on the interval $I$. Then $f$ is uniformly continuous on $I$.
$\large{\text{Satz 2}}:$ If for every $h \gt 0$ is such that $|f(x + h) – f(x)|$ is bounded on $I$, then $f$ is not uniformly continuous on $I$.
Thankful for any help I can get, completely stuck with this. I think it's rather hard to see how to choose the $\delta$'s in particular.
Best Answer
First, we show that $e^{x^2}$ is not uniformly continuous. If we suppose $|x-y| = \delta$, we may choose $x$ and $y$ sufficiently large such that $|x+y| \geq \frac{1}{\delta}$, in which case $|x^2 - y^2| \geq 1$. Then, assuming for convenience $x >y$, $$|e^{x^2} - e^{y^2}| = e^{x^2} (1 - e^{y^2 - x^2}) \geq e^{x^2} \big( 1 - e^{-1} \big)$$ And we see that the above difference can be made as large as we please, no matter which $\delta$ is given. Thus this function is not uniformly continuous.
For the second function, simply note that $e^{-x^2}$ has derivative bounded by $2/e$. Thus by the mean value theorem, choosing $\delta = e \cdot \epsilon /2$, $|x-y| < \delta$ implies $$|e^{-x^2} - e^{-y^2}| \leq \frac{2}{e} \cdot |x-y| < \epsilon$$ A fun generalization for the second problem: suppose $f : \mathbb{R} \to \mathbb{R}$ is continuous and $\displaystyle \lim_{|x| \to \infty} f(x) = 0$. Prove that $f$ is uniformly continuous on $\mathbb{R}$.