1. The problem statement, all variables and given/known data If f(x) be a "continuous" function in interval [a,b] such that f(a)=b and f(b)=a, then prove that there exists at least one value "c" in interval (a,b) such that f(c)=c. Note: [a,b] denotes closed interval from a to b that is a and b inclusive. (a,b) denotes open interval from a to b that is excluding a and b. 2. Relevant equations Concept of continuity. 3. The attempt at a solution As function f(x) is continuous in [a,b] so graph of f(x) between x=a and x=b will be without any "break" and it covers value f(x) from a to b as well. Now as c lies between a and b i.e. a<c<b and f(b)=a and f(a)=b so there should be at least one solution of the equation f(x)=c. But how can we say that solution of equation f(x)=c is x=c ? How can I prove it ? Please help ! Thanks in advanced... :) BTW, coming back after a long time!