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

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 \[\underset{x\to {{a}^{-}}}{\mathop{\lim }}\,f(x)=f(a)=\underset{x\to a+}{\mathop{\lim }}\,f(x)\].