If $f:A\rightarrow B$ is continuous, A is a compact set of some metric $Y$, how dows one show $||f||$ is continuous, ...if it even is continuous. I am inducing that it is since if a real functions is continuous, then its absolute value function is as well.
Continuity of a Sup function. Edited
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real-analysis
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0What is $\sup \|f(x)\|_Y$ defined as a function over? The space $C(A,B)$ of continuous functions from $A$ to $B$? – 2012-04-12
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0the set of x's in A. – 2012-04-12
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1...with the supremum taken over all $x\in A$? – 2012-04-12
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0sorry, Alex, I edited the question. – 2012-04-12
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1@capItan: Do you mean 'show the function $x \mapsto ||f(x)||$ is continuous'? – 2012-04-12
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0@ copper.hat: yes – 2012-04-12
1 Answers
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Using the triangle inequality, you have $||x|| \leq ||x-y|| + ||y||$, and similarly with $x,y$ reversed. From this you can conclude that $|\;||x||-||y||\;| \leq ||x-y||$, ie, the norm is continuous. Now apply this, replacing $x$ by $f(x)$ and $y$ by $f(y)$, and use continuity of $f$.