text: Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 12
  
  # 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.
  
  #### 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?
  
  #### APPEARS IN
  
  RD Sharma Class 12 Maths
  Chapter 9 Continuity
  Exercise 9.1 | Q 39.1 | Page 21