Question

Let [x] denote the integral part of x ∈ R . g(x) = x - [x] . Let f(x) be any continuous function with f(0) = f(1) then the function hex) = f(g(x)) :

(A) has finitely many discontinuities

(B) is discontinuous at some x = c

(C) is continuous on R

(D) is a constant function.

Answer 1

Answer is (C) is continuous on R

g(x) = x - [x] = {x}

f is continuous with f(0) = f(1)

h(x) = f(g(x)) = f(f{x})

Let the graph of f is as shown in the figure

satisfying

f(0) = f(1)

now h(0) = f({0}) = f(0) = f(1)

h(0.2) = f({0.2}) = f(0.2)

h(1.5) = f({1.5}) = f(0.5) etc.

Hence the graph of h(x) will be periodic graph as shown

⇒ h is continuous in R ⇒ C