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Continuity of a Max Function on [0,1] X [0,1]

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Let f(x,y) be a real valued continuous function defined on the unit square [0,1] X [0,1].

Prove g(x)=max{f(x,y) : y in [0,1]} is continuous.

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Can we treat g(x) as a composite function that maps R^2 --> R ?

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Solution Summary

Continuity of a Max Function on [0,1] X [0,1] is investigated.

Solution Preview

Let be a point in [0,1] and let be a sequence in [0,1] converging to .
We want to prove that
We can extend to a sequence in the unit square, converging to ...

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