and Let f and g be two functions given by f(x)= Jo, if z is rational, if z is irrational, 1, if r is rational, [f(x). g(x)={z-f(z), if z is irrational. Prove that, for any real number 1, -|1 ≤ g(x) ≤|z|. Show that the function f is discontinuous at z=0 (in fact, f is discontinuous everywhere on R) and that the function g is continuous at z = 0.

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