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

f(x) = |x – 1| + |x + 1| at x = – 1, 1.

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

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

LHL = f(–1)^{–} =

⇒ |(–2–0)| + |–0|

⇒ |–2|

∴|–x| = |x| = x

⇒ 2 ...(1)

RHL = f(–1) ^{+ =}

⇒ |(–2 + 0)| + |0|

∴|–x| = |x| = x

|–2|

2 ...(2)

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

Hence ,f(x) is continuous at x = –1

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

LHL = f(1)^{–} =

⇒ |–0| + |2–0|

⇒ |2|

⇒ 2 ...(3)

RHL = f(1) ^{+} =

⇒ |0| + |2 + 0|

∴|–x| = |x| = x

⇒ |2|

⇒ 2 ...(4)

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

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

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