Analysis of Continuity and Differentiability

Function:

$$f(x) = \begin{cases} \dfrac{x}{e^{\pi x} - 1}, & x \neq 0 \\ \dfrac{1}{\pi}, & x = 0 \end{cases}$$

✅ Continuity at $x = 0$:

$$\lim_{x \to 0} f(x) = \frac{1}{\pi} = f(0) \quad \Rightarrow \quad \text{Function is continuous at } x = 0$$

✏️ Differentiability at $x = 0$:

$$f'(0) = \lim_{h \to 0} \frac{f(h) - f(0)}{h} = -\frac{1}{2}$$

✅ Final Result:

• Function is continuous at $x = 0$
• Function is differentiable at $x = 0$
• $f'(0) = \boxed{-\frac{1}{2}}$