What is the codomain of A?Define the function: A:C-> as follows: For each f in C, A(f) = the integral from 1 down to 0 f(x)dx.
So, . If , this is the area under the graph of . Since consists of all continuous function, and not only positive ones, it does not matter much that there is a minus in front of ; 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?