Answer to Question #303214 in Real Analysis for Nikhil rawat

Question

Let f be a function defined on R by:

F(x)= { x+5/(x^2-25) , when x≠5

{ 1 when x= 5

Check whether f is uniformly continuous on [-3,3] or not

Answer 1

ANSWER. The function F is uniformly continuous on [-3,3]

EXPLANATION.

$F(x) =\left\{\begin{matrix} \frac{x+5}{ x^{2}-25} ,&if\, x\neq 5 \\ 1,& if x=5 \end{matrix}\right.=\left\{\begin{matrix} \frac{x+5}{ (x-5)(x+5) } ,&if\, x\neq 5 \\ 1,& if x=5 \end{matrix}\right.$ . The function $F$ is defined on the set $D=\left ( -\infty,-5 \right )\cup\left ( -5,+\infty \right )$ . If $x\in[-3,3]$ , then $F(x)=\frac{1 }{x-5}$ and {5} $\notin [-3,3]$ .

Therefore, on the segment [-3,3] the function $F$ is rational and is defined at all points of the segment. So, $F$ is continuous (on [-3,3]). By Cantor's Theorem , a function continuous on a bounded and closed set [-3,3] is a uniformly continuous on this set.