Which one of the following functions is uniformly continuous on the interval (0, 1)?

Concept:

A function y = f(x) is uniformly continuous at an open interval (a, b) if f(x) is continuous on (a, b) and limit exist at end pints a, b.

Explanation:

(1): f(x) = sin

 does not exist so f(x) = sin is not uniformly continuous on (0, 1)

Option (1) is false

(3): f(x) = ex cos

ex cos does not exist so f(x) = ex cos is not uniformly continuous on (0, 1)

Option (3) is false

(4): f(x) = cos x cos

cos x cos does not exist as cos does not exist so f(x) = cos x cos  is not uniformly continuous on (0, 1)

Option (4) is false

(2): f(x) = e−1/x2

(Here f(x) is continuous in (0, 1) and limit exist at x = 0 and x = 1

so f(x) = e−1/x2 is uniformly continuous on (0, 1)

Option (2) is correct