GATE 2022 ST – Question 50

Calculus · Single Variable Calculus · 2 marks · Multiple select

Which real functions are uniformly continuous on [0,∞)?

sin²x
x sin x
sin(sin x)
sin(x sin x)

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Show answer and explanation

Correct answer: (A) sin²x; (C) sin(sin x)

Explanation

The derivatives of A and C are bounded,so both are Lipschitz. For B,take x_n=2πn,y_n=x_n+1/x_n:the distance tends to zero but y_n sin y_n→1 whereas x_n sin x_n=0. For D,use y_n=x_n+π/(2x_n);then y_n sin y_n→π/2,so the outputs tend to 1 and 0. Thus **A,C**.

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