Let $f : R \to R$ be a function such that $f(0) = \frac{1}{\pi}$ and $f(x) = \frac{x}{e^{\pi x} - 1}$ for $x \ne 0$. Then which of the following is correct?

A. $f(x)$ is not continuous at $x = 0$
B. $f(x)$ is continuous but not differentiable at $x = 0$
C. $f(x)$ is differentiable at $x = 0$ and $f'(0) = -\frac{\pi}{2}$
D. None of the above

✅ Correct Answer: $f(x)$ is continuous but not differentiable at $x = 0$

Explanation

Check continuity: $\lim_{x \to 0} \frac{x}{e^{\pi x} - 1} = \frac{1}{\pi} = f(0)$ Hence $f$ is continuous at $x = 0$ Right and left derivatives at $x = 0$ do not exist So $f$ is not differentiable at $x = 0$

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Let $f : R \to R$ be a function such that $f(0) = \frac{1}{\pi}$ and $f(x) = \frac{x}{e^{\pi x} - 1}$ for $x \ne 0$. Then which of the following is correct? | ACME Academy