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 )
Let c is any random number such that c < 1 [thus c being random number, it is able to include all numbers less than 1]
f(c) = [ using eqn 1]
Clearly,
∴ We can say that f(x) is continuous for all x < 1
As x = 1 is a point at which function is changing its nature so we need to check the continuity here.
f(1) = 110 = 1 [using eqn 1]
LHL =
RHL =
Thus LHL = RHL = f(1)
∴ f(x) is continuous at x = 1
Let m is any random number such that m > 1 [thus m being random number, it is able to include all numbers greater than 1]
f(m) = [ using eqn 1]
Clearly,
∴ We can say that f(x) is continuous for all m > 1
Hence, f(x) is continuous for all real x
There no point of discontinuity. It is everywhere continuous