(1) Prove or disprove that there exists functions f.g: RR which are discontinuous at all points p ER, but a function h: R→ R defined by h(x)= (f(x), if xeQ if x € Qº (g(x), is continuous on R. (2) Let f: RR be the function defined by [r³, if req f(x) = x, if x E Q Find all points in R, at which f is continuous. (3) Prove Corollary 5.2.3: For every positive real number y> 0 and every positive integer n, there exists a unique positive real number y such that y" = y. (4) Prove Corollary 5.2.4: If f: [0, 1] → [0, 1] is continuous, then there exists y € [0, 1] such that f(y) = y.