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Question

Show that f(x) = |x − 2| is continuous but not differentiable at x = 2.

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Solution

Given: f(x) = |x-2| = x-2, x2-x+2, x<2

Continuity at x=2: We have,

(LHL at x = 2)
=limx2- f(x) = limh0 f(2-h) = lim h0 (-2+h)+2= 0.

(RHL at x = 2)
=lim x2+f(x) = limh0 f(2+h) = limh0 2+h-2 = 0.

and f(2) = 0

Thus, limx2- f(x) = limx2+ f(x) = f(2)f(2).
Hence, f(x) is continuous at x=2.

Differentiability at x = 2: We have,

(LHD at x = 2)
=limx2- f(x) - f(2)x-2 = limx2 (-x+2) - 0x-2 = limx2 -(x-2)x-2 = limx2 (-1) =-1

(RHD at x=2)
= =limx2+ f(x) - f(2)x-2 = limx2 (x-2) - 0x-2 = limx2 1 = 1

Thus, limx2- f(x) limx2+ f(x).

Hence, f(x) is not differentiable at x=2 .

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