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Question
Discuss the Continuity of the F(X) at the Indicated Points : F(X) = | X − 1 | + | X + 1 | at X = −1, 1.
Solution
Given : \[f\left( x \right) = \left| x - 1 \right| + \left| x + 1 \right|\]
We have
(LHL at x = −1) = \[\lim_{x \to - 1^-} f\left( x \right) = \lim_{h \to 0} f\left( - 1 - h \right)\]
Now,
(LHL at x = 1) = \[\lim_{x \to 1^-} f\left( x \right) = \lim_{h \to 0} f\left( 1 - h \right) = \lim_{h \to 0} \left( \left| 1 - h - 1 \right| + \left| 1 - h + 1 \right| \right) = 0 + 2 = 2\]
(RHL at x =1) = \[\lim_{x \to 1^+} f\left( x \right) = \lim_{h \to 0} f\left( 1 + h \right) = \lim_{h \to 0} \left( \left| 1 + h - 1 \right| + \left| 1 + h + 1 \right| \right) = 0 + 2 = 2\]
Also,
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