# How do you prove that the function f(x) = sqrt(x)  is continuous at 0 to infinity?

Jan 19, 2017

You have to prove that:

${\lim}_{\Delta x \to 0} f \left(x + \Delta x\right) - f \left(x\right) = 0$

#### Explanation:

We have:

$\sqrt{x + \Delta x} - \sqrt{x} = \left(\sqrt{x + \Delta x} - \sqrt{x}\right) \frac{\sqrt{x + \Delta x} + \sqrt{x}}{\sqrt{x + \Delta x} + \sqrt{x}} = \frac{\left(x + \Delta x\right) - x}{\sqrt{x + \Delta x} + \sqrt{x}} = \frac{\Delta x}{\sqrt{x + \Delta x} + \sqrt{x}}$

So that:

${\lim}_{\Delta x \to 0} \sqrt{x + \Delta x} - \sqrt{x} = {\lim}_{\Delta x \to 0} \frac{\Delta x}{\sqrt{x + \Delta x} + \sqrt{x}} = 0$

And the continuity is proved.