Let f; g : D⟶$\mathbb R$ be two continuous functions. f(x) > 0 ∀x∈D. Prove that the function h : D⟶$\mathbb R$ by h(x) = $f(x)^{g(x)}$ is continuous.

Question

I need to prove that : $\lim_{x\to x_0}\ f(x)^{g(x)} = f(x_0) ^{g(x_0)}$

This is what I did: $\lim_{x\to x_0} \ f(x)^{g(x)} =\lim_{x\to x_0} \ e^{ln({f(x)^{g(x)}})}= e^{ln({f(x_0)^{g(x_0)}})}= f(x_0) ^{g(x_0)}$

I really want to know if there is another way to solve this. Thank you!

Answer 1

Hint: We take limit on $\ln h(x) = g(x) \ln f(x)$ instead of $h(x)$.

1. By composition of continuous functions, $\ln \circ f$ is continuous on $D$.
2. The product of two continuous functions $g$ and $\ln \circ f$ are again continuous on $D$.
3. By composition of continuous functions, $\exp \circ (g \cdot \ln \circ f)$ is continuous on $D$. Therefore, $$h(x) = \exp(\ln(h(x))) = \exp(g(x) \ln(f(x)))$$ is continuous.