# JEE Main & Advanced Mathematics Functions Continuity From Left and Right

## Continuity From Left and Right

Category : JEE Main & Advanced

Function $f(x)$ is said to be

(1) Left continuous at $x=a$ if $\underset{x\to {{a}^{-}}}{\mathop{\text{lim}}}\,f(x)=f(a)$

(2) Right continuous at $x=a$ if $\underset{x\to {{a}^{+}}}{\mathop{\text{lim}}}\,f(x)=f(a)$.

Thus a function $f(x)$ is continuous at a point $x=a$ if it is left continuous as well as right continuous at $x=a.$

Properties of continuous functions : Let $f(x)$ and $g(x)$ be two continuous functions at $x=a.$Then

(i) A function $f(x)$ is said to be everywhere continuous if it is continuous on the entire real line R i.e. $(-\infty ,\infty )$. e.g., polynomial function, ${{e}^{x}},$$\sin x,\,\cos x,\,$constant, ${{x}^{n}},$ $|x-a|$ etc.

(ii) Integral function of a continuous function is a continuous function.

(iii) If $g(x)$ is continuous at $x=a$  and $f(x)$  is continuous at $x=g(a)$ then $(fog)\,(x)$ is continuous at $x=a$.

(iv) If $f(x)$ is continuous in a closed interval $[a,\,\,b]$ then it is bounded on this interval.

(v) If $f(x)$ is a continuous function defined on $[a,\,\,b]$ such that $f(a)$ and $f(b)$ are of opposite signs, then there is atleast one value of $x$ for which $f(x)$ vanishes. i.e. if $f(a)>0,\,\,f(b)<0\Rightarrow \,\exists \,\,c\,\,\in \,\,(a,\,\,b)$ such that $f(c)\,=0$.

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