# BIBO Stability of a piecewise function

I have a function $y$ defined as $$y(t) = \left\{\begin{array} ~t~ \mbox{ where} |t| \le 3\\ 0~ \mbox{ otherwise} \end{array} \right .$$ With a system defined as

$$G(t) = ty(t),$$

is it BIBO stable? I know that if my function is defined at all points along the time axis then it will not be stable. But if $t=4, y(t)=0$, would this be considered bounded and thus BIBO stable?

• What is $G(t)$? The statement "if my function is defined at all points along the time axis then it will not be stable" is wrong. – Matt L. Jan 24 '18 at 11:15
• 1. Take definition of BIBO stable from textbook. 2. put in terms you've got. 3. Profit. What exactly do you need help with? – Marcus Müller Jan 24 '18 at 12:21

I'll assume that you are referring to an LTI system with impulse response $G(t)$. The system is stable if the impulse response is absolutely integrable:
$$\int_{-\infty}^{\infty}|G(t)| \ \mathrm{d}t<\infty$$
Because $G(t)$ has a closed form, we can actually compute that integral:
$$\int_{-\infty}^{\infty}|G(t)| \ \mathrm{d}t=\int_{-3}^3|t^2| \ \mathrm{d}t=18 <\infty$$