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Math Help - Let C be the set of all real functions that are continuous on the closed interval [0,

  1. #1
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    Let C be the set of all real functions that are continuous on the closed interval [0,

    I was studying for my final when I stumbled across this question, I am completely stuck and would appreciate any and all help for how you'd go about proving this or finding a counterexample if its false.

    Let C be the set of all real functions that are continuous on the closed interval [0,1]. Define the function: A:C-> as follows: For each f in C, A(f) = the integral from 1 down to 0 f(x)dx.

    Is the function A an injection? Is it a surjection? Justify your conclusions.

    If it is true, or if it is an injection or surjection the justifications should be proofs.
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  2. #2
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    Re: Let C be the set of all real functions that are continuous on the closed interval

    Define the function: A:C-> as follows: For each f in C, A(f) = the integral from 1 down to 0 f(x)dx.
    What is the codomain of A?

    So, A(f)=\int_1^0 f(x)\,dx=\int_0^1(-f(x))\,dx. If -f(x)\ge0, this is the area under the graph of -f(x). Since C consists of all continuous function, and not only positive ones, it does not matter much that there is a minus in front of f; in the first approximation you can think about the area under a continuous line above the x-axis.

    So, do you think that any two lines produce different areas? Think about the area of a triangle and rectangle. Can you make a rectangle with any given area?
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