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f(x)={12, if x is an integer , -7 if x is real number not integer}

find whether f is continuous at x =9

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find whether f is continuous at x =9

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On the real number axis, we have an integer N and its successive integer N+1 or N-1. In between them there are infinite number of real numbers (non-integers). In the neighbourhood of an integer there are only real numbers (non-integers).

So f(9) = 12

f(9-) = -7 as 9 -Δx is a non integer real number as Δx -> 0, but not 0.

f(9+) = -7 for the same reason.

So the function is discontinuous at x = 9 or any integer. It is continuous at all non-integer real numbers.

So f(9) = 12

f(9-) = -7 as 9 -Δx is a non integer real number as Δx -> 0, but not 0.

f(9+) = -7 for the same reason.

So the function is discontinuous at x = 9 or any integer. It is continuous at all non-integer real numbers.