Answer to Question #109863 in Real Analysis for Pappu Kumar Gupta

Question

Check whether the function, f , defined below, is uniformly continuous or not:
f (x) =x^1/2 , x belong to [1,2].

Answer 1

Consider the function $f(x)=\sqrt{x}$

The domain of the function $f(x)=\sqrt{x}$ is $[0,\infty)$

Here, the given interval is $[1,2]$ which lies within the domain of the function $f(x)=\sqrt{x}$

So, the function is uniformly continuous in the interval $[1,2]$ including end points as well.

Therefore, the function $f(x)=\sqrt{x}$ is uniformly continuous in the interval $[1,2]$.

As shown in the graph, the function is continuous in the interval $[0,\infty)$ , so function will be uniformly continuous in any interval lies within this domain.