A real function f is said to be continuous at x = c,
Where c is any point in the domain of f
If :
\(\lim\limits_{h \to 0}f(c-h)\) = \(\lim\limits_{h \to 0}f(c+h)\) = f(c)
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.
A function is continuous at x = c if :
\(\lim\limits_{x \to c}f(x)\) = f(c)
Here we have,
\(f(x) = \begin{cases}\frac{sin\,x}{x}+cos\,x&,\quad if\, x ≠0\\5&,\quad if\,x=0\end{cases} \) …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 numbers except 0 ]
∴ 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 the continuity here.
f(0) = 5 [using eqn 1]
And,
∴ f(x) is discontinuous at x = 0
Hence,
f is continuous for all x ≠ 0 but discontinuous at x = 0.