Let f : [0, ∞) → ℝ be the periodic function of period 1 given by f(x) = 1 − |2x − 1| for x ∈ [0, 1], and define g : [0, ∞) → ℝ by g(x) = f(x²). Which of the following statements are true?

Question

Accepted Answer

A, B, C. f is a continuous tent function with f(0) = f(1) = 0, so its periodic extension is continuous; continuous periodic ⇒ uniformly continuous. g = f∘x² is continuous but its oscillations speed up as x → ∞ (the tents get compressed), so it is not uniformly continuous.

Periodic uniform continuity

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