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\(\rm\frac{1}{x}\)

\(\lim_{x\to0}\sin\frac1x\) does not exist so f(x) = sin\(\rm\frac{1}{x}\) is not uniformly continuous on (0, 1)

Option (1) is false

(3): f(x) = ex cos\(\rm\frac{1}{x}\)

\(\lim_{x\to0}\)ex cos\(\rm\frac{1}{x}\) does not exist so f(x) = ex cos\(\rm\frac{1}{x}\) is not uniformly continuous on (0, 1)

Option (3) is false

(4): f(x) = cos x cos\(\rm\frac{\pi}{x}\)

\(\lim_{x\to0}\)cos x cos\(\rm\frac{\pi}{x}\) does not exist as \(\lim_{x\to0}\)cos\(\rm\frac{\pi}{x}\) does not exist so f(x) = cos x cos\(\rm\frac{\pi}{x}\)  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