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Define $C([a,b], \mathbb R)$ to be the space of continuous functions $f : [a,b] \to \mathbb R$ with the norm $\| \cdot \|_{\infty}$. Let $H : C([a, b], \mathbb{R}) \rightarrow \mathbb{R}$ be the map such that $H(f)$ is the definite integral of $f$ from $a$ to $b$ for any $f \in C([a,b], \mathbb R)$.

How do I prove that $H$ is continuous?

Note [SN]: Edited the question to make it more complete.

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Continuous in $a$? in $b$? in $f$? If you mean continuous in $f$, you need a topology on the function space first. –  Gerry Myerson Nov 4 '11 at 3:02
I guess the $C([a,b], \mathbb R)$ space is endowed with the sup-norm. If so, it might be a good idea to mention that in the question. (Also, I think the usual notation is $C([a,b], \mathbb R)$, not $\mathbb C([a,b], \mathbb R)$.) –  Srivatsan Nov 4 '11 at 3:03
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1 Answer

Yes, this is because it is Lipschitz:

$$ |H(f)-H(g)|=\left|\int_a^b(f-g)\; dx\right|\leqslant |b-a| \cdot \|f-g\|_{\infty} .$$

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+1 I edited the answer slightly; hopefully, it's ok. –  Srivatsan Nov 4 '11 at 3:08
Very grateful actually, I couldn't figure out how to center things, thanks! –  Alex Youcis Nov 4 '11 at 3:09
+1 Changed the english a bit and added the link. –  AD. Nov 4 '11 at 5:17
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