1. The problem is to analyze the function $f(t) = \frac{\sin(t)}{t}$ and understand its behavior. 2. This function is known as the sinc function (unnormalized). It is defined as $f(t) = \frac{\sin(t)}{t}$ for $t \neq 0$. 3. Important rule: At $t=0$, the function is not defined directly because of division by zero. However, using the limit \(\lim_{t \to 0} \frac{\sin(t)}{t} = 1\), we define $f(0) = 1$ to make the function continuous. 4. To analyze the function, note that $\sin(t)$ oscillates between -1 and 1, and dividing by $t$ causes the amplitude to decrease as $|t|$ increases. 5. The function has zeros at all nonzero multiples of $\pi$, i.e., $t = \pm \pi, \pm 2\pi, \pm 3\pi, \ldots$ because $\sin(t) = 0$ at these points. 6. The function has a removable discontinuity at $t=0$ which is resolved by defining $f(0) = 1$. 7. The function is even, since $\frac{\sin(-t)}{-t} = \frac{-\sin(t)}{-t} = \frac{\sin(t)}{t}$. 8. Summary: $f(t) = \frac{\sin(t)}{t}$ for $t \neq 0$, and $f(0) = 1$ by limit. Final answer: The function $f(t) = \frac{\sin(t)}{t}$ with $f(0) = 1$ is continuous and oscillates with decreasing amplitude as $|t|$ increases.