I have some problem to understand the definition of a continuous function in a point.

I have $f(x) = \sqrt{x}$ and I want to check the continuity of the function above in the point $x_0 = 0$ or for all the $x_0 \ge 0$.

Using the relation between $|f(x) -f(x_0)|< \epsilon$ and $|x - x_0| < \delta$ that states: $\forall\epsilon \in \Bbb R, \exists\delta\in\Bbb R : \forall x \in domain \mathcal (f): |x-x_0| < \delta \Rightarrow |f(x)-f(x_0)| < \epsilon$, how can I check continuity at a point?



$|f(x) - f(c)| = |\sqrt{x} - \sqrt{c}| \leq\sqrt{|x-c|}$ this shows the continuity of $\sqrt{x}$

  • $\begingroup$ how should I see the continuity from that? $\endgroup$ – Lc0rE Nov 5 '12 at 16:24
  • 1
    $\begingroup$ if |f(x)-f(c)|< epsilon then your delta will be epsilon^2 $\endgroup$ – jim Nov 5 '12 at 16:29

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