Let’s analyze each function for continuity at $x = 0$, as that is the point where the definition of the functions changes. A function is continuous at $x = 0$ if:

1. The left-hand limit ($LHL$) and right-hand limit ($RHL$) exist and are equal at $x = 0$.
2. The value of the function at $x = 0$ matches these limits.

(A) ( f(x) = \begin{cases}

-x^2 & \text{if } x \leq 0 \ x^2 & \text{if } x > 0 \end{cases} )

• Left-hand limit as $x \to 0^-$: $LHL = -x^2 \to 0$.
• Right-hand limit as $x \to 0^+$: $RHL = x^2 \to 0$.
• Value at $x = 0$: $f(0) = -x^2 = 0$.

Since $LHL = RHL = f(0)$, $f(x)$ is continuous at $x = 0$.

(B) ( f(x) = \begin{cases}

2x + 1 & \text{if } x \leq 0 \ -2x + 1 & \text{if } x > 0 \end{cases} )

• Left-hand limit as $x \to 0^-$: $LHL = 2(0) + 1 = 1$.
• Right-hand limit as $x \to 0^+$: $RHL = -2(0) + 1 = 1$.
• Value at $x = 0$: $f(0) = 2(0) + 1 = 1$.

Since $LHL = RHL = f(0)$, $f(x)$ is continuous at $x = 0$.

(C) ( f(x) = \begin{cases}

1 & \text{if } x \leq 0 \ e^x & \text{if } x > 0 \end{cases} )

• Left-hand limit as $x \to 0^-$: $LHL = 1$.
• Right-hand limit as $x \to 0^+$: $RHL = e^0 = 1$.
• Value at $x = 0$: $f(0) = 1$.

Since $LHL = RHL = f(0)$, $f(x)$ is continuous at $x = 0$.

(D) ( f(x) = \begin{cases}

\sin(x) & \text{if } x \leq 0 \ \cos(x) & \text{if } x > 0 \end{cases} )

• Left-hand limit as $x \to 0^-$: $LHL = \sin(0) = 0$.
• Right-hand limit as $x \to 0^+$: $RHL = \cos(0) = 1$.
• Value at $x = 0$: $f(0) = \sin(0) = 0$.

Since $LHL \neq RHL$, $f(x)$ is not continuous at $x = 0$.

Conclusion

The function (D) is not continuous everywhere.

Would you like detailed steps or further clarifications?

Relative Questions:

1. What are the formal conditions for a function to be continuous at a point?
2. Can a function be continuous everywhere but not differentiable everywhere? Provide examples.
3. How can we determine continuity at endpoints of a domain?
4. What happens if only $LHL = RHL$ but $f(0)$ is undefined?
5. How is piecewise function continuity related to real-world applications?

Tip:

Always check both the limits and the actual function value to ensure continuity.