COMEDK 2025 Afternoon Solved Paper

Concept:The function is discontinuous at a point where the left-hand limit, right-hand limit, and the function value are not equal.Explanation:The function $f(x)=\frac{|x|}{x}$ for $x\neq 0$ can be written as $f(x)=1$ when $x>0$ and $f(x)=-1$ when $x<0$. At $x=0$, $f(0)=0$.Now check continuity at $x=0$:Left-hand limit: $\lim_{x\to 0^-} f(x) = -1$.Right-hand limit: $\lim_{x\to 0^+} f(x) = 1$.Since the left and right limits are different, $\lim_{x\to 0} f(x)$ does not exist.Also, the limits do not equal $f(0)=0$.Thus, $f$ is discontinuous at $x=0$.For any $x>0$ or $x<0$, $f(x)$ is constant, so $f$ is continuous there.Answer:Option A: $x=0$