CUET PG 2023 — Mathematics PYQ

For x < 0, the absolute value $|x|=-x$. Therefore:

Now, we can rewrite the function as:

Step 2: Check Continuity at $x=0$

A function is continuous at $x=a$ if $LHL=RHL=f(a)$.

• Left Hand Limit (LHL):

  $$\underset{x\to 0^{-}}{\lim }f(x)=\underset{x\to 0^{-}}{\lim }(-1)=-1$$

• Right Hand Limit (RHL):

  $$\underset{x\to 0^{+}}{\lim }f(x)=\underset{x\to 0^{+}}{\lim }(-1)=-1$$

• Value of the Function at $x=0$:

  $$f(0)=-1$$

Since $LHL=RHL=f(0)=-1$, the function is continuous at $x=0$.

Step 3: Check for all other points

• For x < 0, $f(x)=-1$, which is a constant function and thus continuous.

• For x > 0, $f(x)=-1$, which is also a constant function and thus continuous.

Final Conclusion

The function $f(x)$ is continuous for all real values of $x$. In interval notation:

$x\in (-\infty ,\infty )$ or $x\in \mathbb{R}$.