# Continuous functions

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1. Apr 28, 2016

1. The problem statement, all variables and given/known data
Prove $$T\int_c^d f(x,y)dy = \int_{c}^dTf(x,y)dy$$ where $$T:\mathcal{C}[a,b] \to \mathcal{C}[a,b]$$ is linear and continuous in L^1 norm on the set of continuous functions on [a,b] and
$$f:[a,b]\times [c,d]$$ is continuous.

2. Relevant equations

3. The attempt at a solution

I couldn't come up with any viable idea. I only know that the integrals are continuous as functions of x.

2. Apr 28, 2016

### andrewkirk

Since $f$ is continuous we know that the Riemann integral exists and is equal to the Lebesgue integral. So re-write the integral as a limit using the Riemann interpretation. It should be easy enough to proceed from there.