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