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## Homework Statement

I have to prove that [itex]\sqrt{x}[/itex] is continuous on the interval [1,[itex]\infty[/itex]).

**2. The attempt at a solution**

Throughout the school semester I believed that to show that a function is continuous everywhere all I need to do was show that [itex]\lim\limits_{h\rightarrow 0}f(x+h)-f(x)=0[/itex] and I never thought much about it. It never really came up as a problem on any of the homeworks or exams so I never had much problem with it. I am now doing review problems for the final and I realized that f(x)=1/x is a clear counterexample to what I stated above. The limit as h approaches 0 of f(x+h)-f(x) is 0 but 1/x is not continuous everywhere. Since this is the case, how do I show that a function is continuous everywhere within an interval?

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