Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 12
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Discuss the Continuity of the F(X) at the Indicated Points: (I) F(X) = | X | + | X − 1 | at X = 0, 1. - Mathematics

Discuss the continuity of the f(x) at the indicated points: 

(i) f(x) = | x | + | x − 1 | at x = 0, 1.

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Solution

Given: 

\[f\left( x \right) = \left| x \right| + \left| x - 1 \right|\]

We have
(LHL at x = 0) = 

\[\lim_{x \to 0^-} f\left( x \right) = \lim_{h \to 0} f\left( 0 - h \right)\]
\[= \lim_{h \to 0} \left[ \left| 0 - h \right| + \left| 0 - h - 1 \right| \right] = 1\]

(RHL at x = 0) = 

\[\lim_{x \to 0^+} f\left( x \right) = \lim_{h \to 0} f\left( 0 + h \right)\]
\[= \lim_{h \to 0} \left[ \left| 0 + h \right| + \left| 0 + h - 1 \right| \right] = 1\]
\[= \lim_{h \to 0} \left[ \left| 0 + h \right| + \left| 0 + h - 1 \right| \right] = 1\]

Also

\[f\left( 0 \right) = \left| 0 \right| + \left| 0 - 1 \right| = 0 + 1 = 1\]

Now,

(LHL at x = 1) =

\[\lim_{x \to 1^-} f\left( x \right) = \lim_{h \to 0} f\left( 1 - h \right) = \lim_{h \to 0} \left( \left| 1 - h \right| + \left| 1 - h - 1 \right| \right) = 1 + 0 = 1\]

(RHL at x =1) = 

\[\lim_{x \to 1^+} f\left( x \right) = \lim_{h \to 0} f\left( 1 + h \right) = \lim_{h \to 0} \left( \left| 1 + h \right| + \left| 1 + h - 1 \right| \right) = 1 + 0 = 1\]

Also,

\[f\left( 1 \right) = \left| 1 \right| + \left| 1 - 1 \right| = 1 + 0 = 1\]
\[f\left( 1 \right) = \left| 1 \right| + \left| 1 - 1 \right| = 1 + 0 = 1\]
\[\lim_{x \to 0^-} f\left( x \right) = \lim_{x \to 0^+} f\left( x \right) = f\left( 0 \right) and \lim_{x \to 1^-} f\left( x \right) = \lim_{x \to 1^+} f\left( x \right) = f\left( 1 \right)\]

Hence,

\[f\left( x \right)\]  is continuous at
\[x = 0, 1\]
  Is there an error in this question or solution?
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APPEARS IN

RD Sharma Class 12 Maths
Chapter 9 Continuity
Exercise 9.1 | Q 39.1 | Page 21
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