The function f is continuous and has the property $ f(f(x)) = 1 - x $ for all x in $ [0, 1] $, and $ J = \int_{0}^{1} f(x) dx $, then find $f(\frac{1}{4}) + f(\frac{3}{4})$ & the value of J.
I have thought a lot about this question, but I haven't really made any significant progress. One thing that I have spotted is that using the property of definite integration, we can reduce the integral to $ J = \int_{0}^{1} f(1 - x) dx $.
Now this is similar to the property of the function. But I can't proceed further. Another line of reasoning was to manipulate the functional equation, by replacing x with 1 - x, so that we get $ f(f(1-x)) = x $. But again, it's a dead end.
