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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2020-04 Convergence at all but one point, 4.0 out of 5 based on 3 ratings

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