f(x)=sqrt x is continuous

**redsoxnation** · February 21st 2010, 03:49 PM

I need help with this proof. I have to use the definition of continuity to prove that the function f defined by f(x)=sqrt x is continuous at every nonnegative real number.

**patrick** · February 21st 2010, 05:19 PM

I'll leave it to you to prove that it's continuous at 0 (it's quite easy).

Suppose x is non-zero and let $\varepsilon >0$ be given. Then, we seek $\delta > 0$ such that

$\left|\sqrt{x}-\sqrt{y}\right| < \varepsilon .$

Use the fact that $\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+\sqrt{y}\right) = x-y$ and the assumption that x is nonzero and you should find the desired $\delta$ quite easily.

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