IGDTUW 2025 — Mathematics PYQ

For a function to be continuous at a point $x=c$, the following condition must be met:

$$\underset{x\to c^{-}}{\lim }f(x)=\underset{x\to c^{+}}{\lim }f(x)=f(c)$$

1. Checking continuity at $x=0$:

• Left-hand limit: ${\lim }_{x\to 0^{-}}f(x)={\lim }_{x\to 0^{-}}(-x)=0$

• Right-hand limit: ${\lim }_{x\to 0^{+}}f(x)={\lim }_{x\to 0^{+}}(x)=0$

• Function value: $f(0)=-0=0$

Since ${\lim }_{x\to 0^{-}}f(x)={\lim }_{x\to 0^{+}}f(x)=f(0)$, the function is continuous at $x=0$.

2. Checking continuity at $x=1$:

• Left-hand limit: ${\lim }_{x\to 1^{-}}f(x)={\lim }_{x\to 1^{-}}(x)=1$

• Right-hand limit: ${\lim }_{x\to 1^{+}}f(x)={\lim }_{x\to 1^{+}}(2-x)=2-1=1$

• Function value: $f(1)=2-1=1$

Since ${\lim }_{x\to 1^{-}}f(x)={\lim }_{x\to 1^{+}}f(x)=f(1)$, the function is continuous at $x=1$.

Conclusion:

The function $f(x)$ is continuous at both $x=0$ and $x=1$. Therefore, the correct option is (a).