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Examine the following functions for continuity. $f(x) = | \;x - 5\;|$

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  • If $f$ is a real function on a subset of the real numbers and $c$ a point in the domain of $f$, then $f$ is continous at $c$ if $\lim\limits_{x\to c} f(x) = f(c)$.
  • Every polynomial function $f(x)$ is continous.
Given $f(x) = | \;x - 5\;|$, $f(x) = x -5$, for $x \geq 5$ and $f(x) = 5-x$ for $x<5$.
$\textbf{Step 1}$:
The function $f$ is defined at all points of the real line.
Every polynomial function $f(x)$ is continous.
At $x >5$, $f(x) = x-5$ and at $x<5$, $f(x) = 5-x$, which are both polynomial functions, and $f(x)$ is continuous in both these cases.
$\textbf{Step 2}$:
At $x=5$, we need to evaluate the Right Hand Limit and Left Hand Limit and see if they are equal, in which case, $f(x)$ is continous.
$\Rightarrow$ RHL: For $x \to 5^+, \lim\limits_{x\to 5^+} f(x) = \lim\limits_{x\to 5^+} x-5 = 5-5 = 0$
$\Rightarrow$ LHL : For $x \to 5^-, \lim\limits_{x\to 5^-} f(x) = \lim\limits_{x\to 5^-} 5-x = 5-5 = 0$
Since RHL = LHL, $f$ is continuous in all cases.
answered Apr 4, 2013 by balaji.thirumalai

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