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 :
Given :
\(f(x) = \begin{cases} x^2sin\frac{1}{x} &, x≠{0}\\ 0 &, x={ 0} \end{cases} \) ….Equation 1
As for x ≠ 0,
f(x) is just a product of two everywhere continuous function
∴ it is continuous for all x ≠ 0.
∵ f(x) is changing its nature at x = 0,
So we need to check continuity at x = 0
f(0) = 0 [using equation 1]
And,
\(\lim\limits_{x \to 0}(x^2sin\frac{1}{x})\) = 0
[∵ sin(1/0) is also going to be a value between [–1,1] ,so its product with 0 = 0]
Thus,
\(\lim\limits_{x \to 0}f(x)\) = f(0)
∴ It is continuous at x = 0
Hence, it is everywhere continuous.
