Is the function defined by f(x) = | x |, a continuous function? - Vidyadip

Question

Is the function defined by f(x) = | x |, a continuous function?

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Answer

We can write f as
$f(x)=\left\{\begin{array}{ll} {-x,} & {\text { if } x<0} \\ {~~~x,} & {\text { if } x \geq 0} \end{array}\right.$
we know that f is continuous at x = 0.
Suppose c be a real number such that c < 0. Then f(c) = – c. Also
$\mathop {\lim }\limits_{x \to c} f(x) = \mathop {\lim }\limits_{x \to c} $ (-x) = -c
Since $\mathop {\lim }\limits_{x \to c} f(x) $ = f(c), f is continuous at all negative real numbers.
Now, suppose c be a real number such that c > 0. Then f(c) = c. Also
$\mathop {\lim }\limits_{x \to c} f(x) = \mathop {\lim }\limits_{x \to c} $x = c
Since $\mathop {\lim }\limits_{x \to c} f(x)$ = f(c), f is continuous at all positive real numbers. Therefore, f is continuous at all points.

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