**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).

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

*play_arrow*Some Important Definitions*play_arrow*Intervals*play_arrow*Definition of Function*play_arrow*Domain, Co-domain and Range of Function*play_arrow*Algebra of Functions*play_arrow*Kinds of function*play_arrow*Even and Odd Function*play_arrow*Periodic Function*play_arrow*Composite Function*play_arrow*Inverse Function*play_arrow*Limit of a Function*play_arrow*Fundamental Theorems on Limits*play_arrow*Methods of Evaluation of Limits*play_arrow*Introduction*play_arrow*Continuity of a Function at a Point*play_arrow*Continuity From Left and Right*play_arrow*Discontinuous Function*play_arrow*Differentiability of a Function at a Point

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