Q. Function $f(x) = \left\{ \begin{array}{r} x - 1,\ x < 2 \\ 2x - 3,\ x \geq 2 \end{array} \right.\$ is a continuous function
A) For $x = 2$ only
B) For all real values of $x$ such that $x \neq 2$
C) For all real values of $x$
D) For all integer values of $x$ only
Explanation
(c)
Since, it is a polynomial function, so it is continuous for every value of $x$ except at $x = 2$
LHL$= \lim_{x \rightarrow 2^{-}}{x - 1}$
$= \lim_{h \rightarrow 0}\ 2 - h - 1 = 1$
RHL$= \lim_{x \rightarrow 2^{\mp}}2x - 3 = \lim_{h \rightarrow 0}\ 2(2 + h) - 3 = 1$
And $f(2) = 2(2) - 3 = 1$
$\therefore$ LHL+RHL$= f(2)$
Hence, $f(x)$ is continuous for all real values of $x$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.