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

f(x) = |x| + |x – 1| at x = 0, 1.

To prove whether f(x) is continuous at x = 0 & 1

If f(x) to be continuous at x = 0,we have to show that f(0)^{–}=f(0) ^{+} = f(0)

LHL = f(0)^{–} =

⇒ |(–0)| + |(–0)–1|

∴–|x| = |x| = x

⇒ |–1|

⇒ 1 ...(1)

RHL = f(0) ^{+} =

⇒ |0| + |0–1|

∴–|x| = |x| = x

⇒ |–1|

⇒ 1 ...(2)

From (1) & (2),we get f(0)^{–}=f(0) ^{+}

Hence ,f(x) is continuous at x = 0

If f(x) to be continuous at x = 1,we have to show, f(1)^{–} =f(1) ^{+} = f(1)

LHL = f(1)^{–} _{=}

⇒ |(1–0) + (–0)|

⇒ |(1)|

⇒ 1 ...(3)

RHL = f(1) ^{+ =}

⇒ |(1 + 0)| + |0|

∴–|x| = |x| = x

⇒ |1|

⇒ 1 ...(4)

From (3) & (4),we get f(1)^{–} _{=} f(1) ^{+}

Hence ,f(x) is continuous at x = 1

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