JEE Main & Advanced Mathematics Functions Continuity From Left and Right

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 \[f(c)\,=0\].