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The function f(x) is bounded due to its continuity on the compact interval [0, 1]. This is a valid conclusion from the fact that f is continuous and f(x) = f(x + 1) for all x ∈ ℝ, which implies the periodicity of the function with a period of 1.
[IIT JAM 2021]

a function is periodic with period 1 if its values repeat every 1 unit along the x-axis. This means that the function's behavior within any 1-unit interval is the same as in any other 1-unit interval[IIT JAM 2021]