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Continuity and Differentiability — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsContinuity and Differentiability2 Marks
Question
Discuss the continuity of the function defined by
$f(x)=\left\{\begin{array}{c} {x+2, \text { if } x<0} \\ {-x+2, \text { if } x>0} \end{array}\right.$
✓

Answer

Observe that the function is defined at all real numbers except at 0. The domain of definition of this function is
D1 $\cup$ D2 where D1 = {x $\in$ R : x < 0} and D2 = {x $\in$ R : x > 0}
Case 1: If c $\in$ D1, then $\mathop {\lim }\limits_{x \to c} f(x) = \mathop {\lim }\limits_{x \to c} \,\,$(x + 2)
= c + 2 = f (c) and hence f is continuous in D1.
Case 2: If c $\in$ D2, then $\mathop {\lim }\limits_{x \to c} f(x) = \mathop {\lim }\limits_{x \to c} \,\,$(-x + 2)
= – c + 2 = f (c)

and hence f is continuous in D2. Since f is continuous at all points in the domain of f, we deduce that f is continuous. The graph of this function is given in the figure. Note that to graph this function we need to lift the pen from the plane of the paper, but we need to do that only for those points where the function is not defined.

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