Let f1, . . . , fn be real valued functions defined on a set S in R. Assume that each fi is continuous at a ∈ S. For each x ∈ S, define f(x) to be the largest of the numbers f1(x), . . . , fn(x). Is f continuous at a?
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1Since $\max(f,g)=\frac{1}{2}\left[(f+g)+|f-g|\right]$. The right side is a sum of continuous functions, so max function is continuous. – Anurag A Oct 28 '18 at 05:41
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Extension to any natural number n? – Todd Oct 28 '18 at 05:45
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2Hint: $\max(f,g,h)=\max(f,\max(g,h))$. – Anurag A Oct 28 '18 at 06:03