Could someone please suggest a good way to check for continuity (using the epsilon-delta definition or maybe using some other method.
An example is to show that $f(x)=\sqrt x$ i.e. $f(x)=x^{1/2}$ is uniformly continuous on the positive real numbers.
Thank you! Your help will be greatly appreciated.
If $x,y > 0$, then $\vert\sqrt{x}-\sqrt{y}|^{2} \leq \vert\sqrt{x}-\sqrt{y}\vert\cdot\vert\sqrt{x}+\sqrt{y}\vert = |x-y|;\;$ given any $\varepsilon > 0$, we have $\vert\sqrt{x}-\sqrt{y}\vert \lt \varepsilon,$ if in addition we have $\vert x-y\vert \lt \varepsilon^{2}.\;$ So taking $\delta := \varepsilon^{2}$ suffices.
Just in case you are not yet that familiar with an epsilon argument:
What the above says is that for every $\varepsilon \gt 0,$ there is some $\delta \gt 0,\;$ say $\delta := \varepsilon^{2}$, such that $\vert x-y\vert \lt \delta$ implies $\vert\sqrt{x}-\sqrt{y}\vert < \varepsilon$.