I have the following function
$$ f(x) = \begin{cases} x & \text{if } x \le 1 \\ x^2 & \text{if }x>1 \end{cases} $$
How do I show that this function is continuous at $x_0=1$?
Continuity of a piecewise function.
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real-analysis
analysis
continuity
epsilon-delta
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0I have tried to use the epsilon delta definition but I am not sure which condition to use. – 2017-01-16
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1Show that the two limits, from left and right, for $x \to 1$ exist and have the same value. – 2017-01-16
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3Hint: the continuity at a point $x_0$ satisfies three conditions: (1) $f(x_0)$ is defined; (2) $\lim_{x \rightarrow x_0} f(x) $ exists, and (3) $\lim_{x \rightarrow x_0} f(x) = f(x_0).$ – 2017-01-16
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0Could I also use epsilon delta proof. If I can, do I have to show for both cases then? – 2017-01-16