Find the points of discontinuity, if any, of the following functions :

Basic Idea:


A real function f is said to be continuous at x = c, where c is any point in the domain of f if :


where h is a very small ‘+ve’ no.


i.e. left hand limit as x c (LHL) = right hand limit as x c (RHL) = value of function at x = c.


This is very precise, using our fundamental idea of limit from class 11 we can summarise it as, A function is continuous at x = c if :



Here we have,


……….equation 1


Function is defined for all real numbers so we need to comment about its continuity for all numbers in its domain (domain = set of numbers for which f is defined)


For x < –1, f(x) is having a constant value, so the curve is going to be straight line parallel to x–axis.


So, it is everywhere continuous for x < –1.


It can be verified using limits as discussed in previous problems


Similarly for –1 < x < 1, plot on X–Y plane is a straight line passing through origin.


So, it is everywhere continuous for –1 < x < 1.


And similarly for x > 1, plot is going to be again a straight line parallel to x–axis


it is also everywhere continuous for x > 1



From graph it is clear that function is continuous everywhere but let’s verify it with limits also.


As x = –1 is a point at which function is changing its nature so we need to check the continuity here.


f(–1) = –2 [using eqn 1]


LHL =


RHL =


Thus LHL = RHL = f(–1)


f(x) is continuous at x = –1


Also at x = 1 function is changing its nature so we need to check the continuity here too.


f(1) = 2 [using eqn 1]


LHL =


RHL =


Thus LHL = RHL = f(1)


f(x) is continuous at x = 1


Thus, f(x) is continuous everywhere and there is no point of discontinuity.


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