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

The 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 < 0 [thus c being a random number, it can include all negative numbers ]

f(c) = [ using eqn 1]

Clearly,

∴ We can say that f(x) is continuous for all x < 0

Now, let m be any random number from the domain of f such that m > 0

thus m being a random number, it can include all positive numbers]

f(m) = 2m + 3 [using eqn 1]

Clearly,

∴ We can say that f(x) is continuous for all x > 0

As zero is a point at which function is changing its nature so we need to check LHL, RHL separately

f(0) = 2×0+3 = 3 [using eqn 1]

LHL =

[∵ sin –θ = – sin θ and ]

RHL =

Thus LHL ≠ RHL

∴ f(x) is discontinuous at x = 0

Hence, f is continuous for all x ≠ 0 but discontinuous at x = 0.