Answer to Question #175914 in Real Analysis for Nikhil Singh

Question

The function: f:[-1,3] →R defined by:

f(x)= 3x+1/x^2+4 is uniformly continuous on [-1,3].

True or false with full explanation

Answer 1

Let us show that the function $f: [-1,3] \to \mathbb R,\ f(x)=\frac{ 3x+1}{x^2+4},$ is uniformly continuous on $[-1,3]$.

Since the elementary functions $g(x)=3x+1$ and $h(x)=x^2+1$ are continuous on $[-1,3]$, and $h(x)>0$, we conclude that the function $f(x)=\frac{g(x)}{h(x)}$ is also continuous on $[-1,3].$ The Heine-Cantor theorem asserts that every continuous function on a compact set is uniformly continuous. Since the interval $[-1,3]$ is closed and hence compact, we conclude that $f$ is uniformly continuous on $[-1,3]$.

Answer: true