For the function $f(x) =\left\{\begin{matrix}\frac{1-x}{|x-1|}&;x<1\\1&;x=1\\x^2&;x>1\end{matrix}\right.$ which of the following is true

it is continuous at all points

The correct answer is Option (1) → it is continuous at all points

$f(x) =\left\{\begin{matrix}\frac{1-x}{|x-1|}&;x<1\\1&;x=1\\x^2&;x>1\end{matrix}\right.$

we know, $|x-1|=1-x$ for $x<1$

$⇒f(x) =\left\{\begin{matrix}1&;x<1\\1&;x=1\\x^2&;x>1\end{matrix}\right.$

$LHL=\lim\limits_{x→1^-}f(x)=1$

$f(1)=1$

$\lim\limits_{x→1^+}f(x)=(1)^2=1$

$LHL=RHL=f(1)$

f is continuous for every x