# Prove Continuous Functions Homework: T Integral from c to d

In summary, the problem asks us to prove the equality between two integrals involving a linear and continuous transformation T and a continuous function f. We can use the fact that f is continuous to rewrite the integrals as a limit using the Riemann interpretation, which will allow us to proceed with the proof.

## Homework Statement

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.

## The Attempt at a Solution

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

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.

## 1. What is the definition of a continuous function?

A continuous function is a function in which the value of the function at a given point is very close to the value of the function at any point in its nearby neighborhood. In other words, the graph of a continuous function has no breaks or gaps.

## 2. How is continuity of a function proved using the T integral?

The continuity of a function can be proved using the T integral by evaluating the integral from the starting point (c) to the ending point (d) and showing that the value of the integral is equal to the change in the function at those points. If the value of the integral and the change in the function are equal, then the function is continuous.

## 3. What are the conditions for a function to be continuous using the T integral?

The conditions for a function to be continuous using the T integral are that the function must be defined on a closed interval [c, d], the function must be integrable on that interval, and the limit of the integral from c to d must exist and be equal to the change in the function at those points.

## 4. Can the T integral be used to prove continuity of all types of functions?

Yes, the T integral can be used to prove continuity of all types of functions as long as the function meets the conditions for continuity mentioned in the previous question. This includes polynomial, rational, trigonometric, and exponential functions.

## 5. How is the T integral different from other methods of proving continuity?

The T integral is different from other methods of proving continuity because it directly relates the continuity of a function to the concept of integration. Other methods, such as the epsilon-delta definition of continuity, are more abstract and do not involve integration. Additionally, the T integral is particularly useful for showing continuity of piecewise functions, which may be difficult to prove using other methods.