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Suppose $f_n$ converges uniformly to a function $f$. Both the sequence and the function are continuous. Moreover, $f$ is strictly convex on $(0,\delta)$ for some arbitrarily small $\delta$. Is it the case that for some $N$, $f_n$ is strictly convex on "almost everywhere" $(0,\delta)$ for $n>N$ if each $f_n$ is convex fucntion on $\mathbb{R}$.

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