# Solution: 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 $$\lim_{n \to \infty} g_n(y_0) = g(y_0)$$.

The best solution was submitted by 홍의천 (수리과학과 2017학번). Congratulations!

Here is his solution of problem 2020-04.

Other solutions were submitted by 길현준 (수리과학과 2018학번, +3), 김기택 (수리과학과 2015학번, +3), 이준호 (2016학번, +3).

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