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,
\(f(x) = \begin{cases} \frac{sin\,x}{x}&, \quad \text{if } x <{0}\\ x+1& ,\quad \text{if } x≥{ 0} \end{cases} \) …….equation 1
To prove it everywhere continuous we need to show that at every point in domain of f(x)
[domain is nothing but a set of real numbers for which function is defined]
Where c is any random point from domain of f.
Clearly from definition of f(x) { see from equation 1},
f(x) is defined for all real numbers.
∴ we need to check continuity for all real numbers.
Let c is any random number such that c < 0
[thus c being a random number, it can include all negative numbers ]
∴ We can say that f(x) is continuous for all x < 0
Now,
Let m be any random number from domain of f such that m > 0
Thus,
m being a random number, it can include all positive numbers.
f(m) = m+1 [using eqn 1]
∴ 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) = 0+1 = 1 [using eqn 1]
Thus,
LHL = RHL = f(0).
∴ f(x) is continuous at x = 0
Hence,
We proved that f is continuous for x < 0 ;
x > 0 and x = 0
Thus,
f(x) is continuous everywhere.
Hence, proved.