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Question 6:
Find all points of discontinuity of f, where f is defined by
The given function f is
It is evident that the given function f is defined at all the points of the real line.
Let c be a point on the real line. Then, three cases arise.
(i) c < 2
(ii) c > 2
(iii) c = 2
Case (i) c < 2
Therefore, f is continuous at all points x, such that x < 2
Case (ii) c > 2
Therefore, f is continuous at all points x, such that x > 2
Case (iii) c = 2
Then, the left hand limit of f at x = 2 is,
The right hand limit of f at x = 2 is,
It is observed that the left and right hand limit of f at x = 2 do not coincide.
Therefore, f is not continuous at x = 2
Hence, x = 2 is the only point of discontinuity of f.
