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

Correct option is B

Analysis of Options:

Option (a): f(x) = sin(1/x)

• As $x\to 0$^$0+$, $1/x\to \infty$, causing the oscillations of $\sin (1/x)$ to become arbitrarily fast.
• Uniform continuity fails because the function does not maintain a uniform "rate of change" as $x$ approaches $0$.
• Not uniformly continuous.

​$$(b) \quad \lim_{x \to 0^+} e^{-\frac{1}{x^2}} = 0 \quad \& \quad \lim_{x \to 1^-} e^{-\frac{1}{x^2}} = e^{-1} = \frac{1}{e} \\\text{So }\\ e^{-\frac{1}{x^2}} \text{ is Uniformly Continuous in } (0,1)\\\ \\(c) \quad \because \lim_{x \to 0^+} e^x \cos\frac{1}{x} \text{ does not exist, so } e^x \cos\frac{1}{x} \text{ is not Uniformly Continuous in } (0,1)\\\ \\(d) \quad \because \lim_{x \to 0^+} \cos x \cos\frac{\pi}{x} \text{ does not exist, so } \cos x \cos\frac{\pi}{x} \text{ is not Uniformly Continuous in } (0,1)$$​​