Question

f(x)={12, if x is an integer , -7 if x is real number not integer}
find whether f is continuous at x =9

Answer 1

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.