Consider the function f: (0, ∞) → R defined by f(x) = e ^ - | _ e...

We know that, $\left|\log x\right|$ is continuous in \left(0, \infty\right)$.</p><p>$⇒ f\left(x\right) = {e}^{- \log \left|x\right|} is continuous in \left(0, \infty\right)$</p><p>Thus, the number of points where$f\left(x\right)$is discontinuous is$0$.</p><p>$⇒ m = 0$</p><p>$f\left(x\right) = \left\{$\begin{cases}{e}^{\log x}, & 0 < x < 1 \\ {e}^{- \log x}, & x \ge 1\end{cases}$\right)

⇒ f\left(x\right) = \left\{\begin{cases}x, & 0 < x < 1 \\ \dfrac{1}{x}, & x \ge 1\end{cases}\right)</p><p>⇒ {f}^{'}\left(x\right) = \left\{$\begin{cases}1, & 0 < x < 1 \\ \dfrac{- 1}{{x}^{2}}, & x \ge 1\end{cases}$\right)

$⇒ {f}^{'}\left({1}^{-}\right) = 1 \text{ and } {f}^{'}\left({1}^{+}\right) = - 1$

So, the number of points where $f\left(x\right)$ is non-differentiable is $1$.

$⇒ n = 1$

$⇒ m + n = 1$