Discuss the continuity of the function f given by f(x) = | x | at… - Vidyadip

Question

Discuss the continuity of the function f given by f(x) = | x | at x = 0.

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Answer

By definition
f(x) = $\left\{\begin{array}{l} {-x, \text { if } x<0} \\ {~~~x, \text { if } x \geq 0} \end{array}\right.$
Clearly the function is defined at 0 and f(0) = 0.
Left-hand limit of f at 0 is
$\mathop {\lim }\limits_{x \to 0^-} f(x) = \mathop {\lim }\limits_{x \to 0^-} $ (-x) = 0
Similarly, the right-hand limit of f at 0 is
$\mathop {\lim }\limits_{x \to 0^+} f(x) = \mathop {\lim }\limits_{x \to 0^+} $ (x) = 0
Thus, the left-hand limit, right-hand limit and the value of the function coincide at x = 0. Hence, f is continuous at x = 0.

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