1. ## continuity

Use the definition of continuity to prove that the function f defined by f(x) = sqrt(x) is continuous at every nonnegative real number.

thanks!

2. Let $(a,b)$ be an open interval in $D = [0, \infty$. Then

$f^{-1}\left((a,b)\right) = \{x \in D : f(x) \in (a,b)\} = \{x \in D : a < \sqrt x < b\}$

$= \{x \in D : a^2 < x < b^2\} = (a^2,b^2)$.

Since the inverse image of every open interval is an open interval, $f$ is continuous on $D$.