Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
    
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
2  
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
add comment

1 Answer

Yes, this is because it is Lipschitz:

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

share|improve this answer
1  
+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
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.