# JEE Main & Advanced Mathematics Functions Continuity of a Function at a Point

## Continuity of a Function at a Point

Category : JEE Main & Advanced

A function $f(x)$ is said to be continuous at a point $x=a$ of its domain if and only if it satisfies the following three conditions :

(1) $f(a)$ exists. ($'a'$ lies in the domain of $f$)

(2) $\underset{x\to a}{\mathop{\lim }}\,\,f(x)$ exist i.e.$\underset{x\to {{a}^{+}}}{\mathop{\lim }}\,f(x)=\underset{x\to {{a}^{-}}}{\mathop{\lim }}\,f(x)$ or R.H.L. = L.H.L.

(3) $\underset{x\to a}{\mathop{\lim }}\,f(x)=f(a)$ (limit equals the value of function).

Cauchy’s definition of continuity : A function $f$ is said to be continuous at a point $a$ of its domain $D$ if for every $\varepsilon >0$ there exists $\delta >0$ (dependent on $\varepsilon )$ such that $|x-a|<\delta$ $\Rightarrow |\,f(x)-f(a)|<\varepsilon .$

Comparing this definition with the definition of limit we find that $f(x)$ is continuous at $x=a$ if $\underset{x\to a}{\mathop{\lim }}\,f(x)$ exists and is equal to $f(a)$ i.e., if $\underset{x\to {{a}^{-}}}{\mathop{\lim }}\,f(x)=f(a)=\underset{x\to a+}{\mathop{\lim }}\,f(x)$.

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