23 questions linked to/from $\sqrt x$ is uniformly continuous
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### Is $\sqrt{x}$ uniformly continuous on $(0,1)$? [duplicate]

Is $\sqrt{x}$ uniformly continuous on $(0,1)$? And can anybody give me a function that is uniformly continuous on some interval but $f^2$ is not?
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### Techniques to prove a function is uniformly continuous

So I understand the definition of uniform continuity, but wanted some suggestions to prove that a function is or isn't uniformly continuous. I have looked ahead and have seen that if a function is ...
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### Strictly increasing continuous function

Prove that any onto strictly increasing map $f: (0,1) \to (0,1)$ is continuous. Since its strictly increasing then for $x<y$ it implies that $f(x) < f(y)$. For continuity I must show that for ...
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### Direct proof. Square root function uniformly continuous on $[0, \infty)$ (S.A. pp 119 4.4.8)

(http://math.stanford.edu/~ksound/Math171S10/Hw8Sol_171.pdf) Prove for all $e > 0,$ there exists $d > 0$ : for all $x, y \ge 0$, $|x - y| < d \implies |\sqrt{x} - \sqrt{y}| < e$. (a) ...
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### Is $1 / \sqrt{x}$ Riemann integrable on $[0,1]$?

If $\int_0^1 1 / \sqrt{x}$ Riemann integrable then using second fundamental theorem of calculus i can easily say that $\sqrt{x}$ is uniformly continuous. Basically it has one point i.e $0$ where it ...
### How to prove that $\sqrt x$ is continuous in $[0,\infty)$?
I am trying to prove that $\sqrt x$ is continuous in $[0,\infty)$. I have started writing the following proof: Given $x_0 \in [0,\infty)$ and $\epsilon > 0$. We have to show that there exists ...
### Proving $f(x)=\sqrt{x}$ is uniformly continuous.
Where did I go wrong? The text I'm using gave a different proof. Help. Attempt: Given any $\epsilon>0$. Let $\delta=\epsilon$ so that if $\vert x-y\vert<\delta$, then \begin{align} \vert x-y\...