Let's go through each part of this problem regarding the Heaviside function $H(t)$.

Part (a): Sketch the graphs of $H(t)$, $H(-t)$, and $H(t - a)$.

1. $H(t)$:

   • The function $H(t)$ is defined as: 1, & t \geq 0 \\ 0, & t < 0 \end{cases}$$
   • This function "jumps" from 0 to 1 at $t = 0$.
   • Graph of $H(t)$: A step function that stays at 0 for $t < 0$ and becomes 1 for $t \geq 0$.

2. $H(-t)$:

   • For $H(-t)$, we replace $t$ with $-t$, resulting in: 1, & t \leq 0 \\ 0, & t > 0 \end{cases}$$
   • Graph of $H(-t)$: This is a mirror image of $H(t)$ about the vertical axis $t = 0$. It stays at 1 for $t \leq 0$ and becomes 0 for $t > 0$.

3. $H(t - a)$:

   • The function $H(t - a)$ represents a shift of $H(t)$ by $a$ units to the right. 1, & t \geq a \\ 0, & t < a \end{cases}$$
   • Graph of $H(t - a)$: This is the same step function but shifted rightward by $a$. The "jump" now occurs at $t = a$, so it stays 0 for $t < a$ and becomes 1 for $t \geq a$.

Part (b): Check the continuity of $H(t)$ and justify your answer.

The Heaviside function $H(t)$ is discontinuous at $t = 0$ because it "jumps" from 0 to 1 at that point. Continuity requires that the left-hand limit and the right-hand limit at a point be equal and match the function’s value at that point. Here:

• As $t$ approaches 0 from the left, $H(t)$ approaches 0.
• As $t$ approaches 0 from the right, $H(t)$ approaches 1.

Since the left-hand limit $\lim_{t \to 0^-} H(t) = 0$ and the right-hand limit $\lim_{t \to 0^+} H(t) = 1$ do not match, $H(t)$ is not continuous at $t = 0$.

Would you like further details or have any questions about this explanation?

Additional Questions for Deeper Understanding:

1. How is the Heaviside function used in the Laplace Transform for solving differential equations?
2. What other types of step functions are commonly used in mathematical modeling?
3. Can the Heaviside function be defined in terms of other elementary functions?
4. What are some practical applications of the Heaviside function in engineering?
5. How does the Dirac delta function relate to the derivative of the Heaviside function?

Tip:

When dealing with piecewise functions, carefully examine each segment's definition and understand its behavior at the boundaries to determine continuity and differentiability.