# Uniform Continuity of a Function on the Unbounded Interval $[a,\infty)$

Problem: Prove or disprove that the function $f(x) := m x^{m} \cdot \sin \left( \dfrac{1}{x^{n}} \right)$ defined on $[a,\infty)$, where $a > 0$ and $m > n > 0$, is uniformly continuous on $[a,\infty)$.

Clearly, $f$ is uniformly continuous on the bounded closed interval $[a,b]$ for any $b > a$, but the question is about the uniform continuity of $f$ on the unbounded interval $[a,\infty)$.

• $\frac{y}{2} \lt \sin(y) \lt y$ for $0 \lt y \lt 1$
• let $y = \dfrac{1}{x^n}$
• remember $m \gt n$