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Question

Show that fx=x-x2, whenx0 2 , whenx=0
is discontinuous at x = 0.

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Solution

The given function can be rewritten as:
fx=x-x2, when x>0x+x2, when x<02, when x=0
fx=0, when x>0x, when x<02, when x=0

We observe
(LHL at x = 0) = limx0-fx=limh0f0-h=limh0f-h=limh0-h=0


(RHL at x = 0) = limx0+fx=limh0f0-h=limh0fh= limh00=0

And, f0=2
∴ ​limx0-fx =limx0+fx f0

Thus, f(x) is discontinuous at x = 0
.


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