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प्रश्न
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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उत्तर
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
\[x = 0, 1\]
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