# 2020-04 Convergence at all but one point

Let $$f_n : [-1, 1] \to \mathbb{R}$$ be a continuous function for $$n = 1, 2, 3, \dots$$. Define
$g_n(y) := \log \int_{-1}^1 e^{y f_n(x)} dx.$
Suppose there exists a continuous function $$g: \mathbb{R} \to \mathbb{R}$$ and $$y_0 \in \mathbb{R}$$ such that $$\lim_{n \to \infty} g_n(y) = g(y)$$ for all $$y \neq y_0$$. Prove or disprove that $$\lim_{n \to \infty} g_n(y_0) = g(y_0)$$.

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