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I have to solve this problem:

Suppose that f is a continuous, vector-valued function and that T is a linear transformation on R^3. Show that "T o f" (stands for T composite f) is continuous and that INTEGRAL[T o f] = T(INTEGRAL[f]).

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I think i know the general epsilon - delta proof for continuity of composite functions. The theorem states that "If f:R^n -> R^m is continuous at vector "a" and T:R^m -> R^p is continuous at f(a) then Tof:R^n ->R^p is continuous at vector a."

The proof is as follows:

For any E>0, we want to show that there exists a d>0 such that |x-a|<d => |T(f(x)) - T(f(a)| < E.

Since T is continuous at f(a), then for E>0, there exists a d1 such that |f(x)- f(a)| < d1 => |T(f(x))-T(f(a))| < E

For f, use d1 as its "E value". Then there exists d2 > 0 such that |x-a|< d2 => |f(x)-f(a)|< d1.

Let d=d2. then:

|x-a| < d => |f(x) - f(a)| < d1

but from the continuity of T, we know |f(x)-f(a)| < d1 => |T(f(x))-T(f(a))| < E

so |x-a| < d => |f(x) - f(a)| < d1 => |T(f(x))-T(f(a))| < E

which proves that Tof is continuous at a.

however, this proof only works if both f and T are continuous. going back to the original question, it states that f is continuous, but it doesn't say anything about the continuity of T. so how do i find whether T is continuous or not? I was just wondering, are all linear transformations continuous? also, what is the significance of the statement "T is a linear transformation on R^3" - is the fact that it's "R^3" something i need to use in my proof? i must admit i'm not very familiar with linear transformations.

also, the proof i wrote above is for continuity at a specific point a. the question just states that f is continuous - i'm assuming it's continuous everywhere. would i still use the same proof for this question?

I don't really know how to do the second part of the problem: show that INTEGRAL[T o f] = T(INTEGRAL[f]). i suspect that it has something to do with the chain rule but i only know how to use the chain rule for derivatives. how do you deal with composite functions when it comes to integrals?

any help would be great - thanks in advance!