I need help in proving those problems.
Q1) Suppose f: [a,b] ->R is continuous and that f([a,b]) is subset of Q. Prove that f is constant on [a,b] using Intermediate value Theorem.
Q2) Prove that 2^x = 3x for some x E (o,1).
Q3) Let f(x) = (x^2 - 4x - 5) / (x-5) for x not equal to 5. Can f(5) be defined so as to make f continuous at 5.