Argue that the impromper integral $\int_1^\infty \frac{1-e^{-2x^2}}{x^2}$ for $x \neq 0$ is convergent
Argue that there exists a continuous function $g: \mathbb{R} \rightarrow \mathbb{R}$ which satisfies $g(x) = \frac{1-e^{-2x^2}}{x^2}$ for $x \neq 0$