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,
We can summarise it as,
A function is continuous at x = c if :
Here we have,
\(f(x) = \begin{cases} 2x-1 &, \quad \text{if } x <{2}\\ \frac{3x}{2} & ,\quad \text{if } x≥{ 2} \end{cases} \) …….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)
Function is changing its nature (or expression) at x = 2,
So we need to check its continuity at x = 2 first.
[using eqn 1]
Clearly,
LHL = RHL = f(2)
∴ function is continuous at x = 2
Let c be any real number such that c > 2
∴ f(c) = \(\frac{3c}{2}\) [using eqn 1]
And,
∴ f(x) is continuous everywhere for x > 2.
Let m be any real number such that m < 2
∴ f(m) = 2m - 1 [using eqn 1]
∴ f(x) is continuous everywhere for x < 2.
Hence,
We can conclude by stating that f(x) is continuous for all Real numbers
