7. Rewrite the following piecewise continuous function in terms of unit step functions \[ f(t)=\left\{\begin{array}{l} 0,0 \leq t<1 \\ t-1,1 \leq t<2 \\ t^{2}, t \geq 2 \end{array}\right. \]

Step 1 ：The given piecewise function is \(f(t)=\left\{\begin{array}{l} 0,0 \leq t<1 \\ t-1,1 \leq t<2 \\ t^{2}, t \geq 2 \end{array}\right.\)

Step 2 ：We can represent this function in terms of unit step functions. The unit step function, also known as the Heaviside function, is a discontinuous function whose value is zero for negative arguments and one for positive arguments.

Step 3 ：For the interval 0 <= t < 1, the function value is 0. This can be represented by the unit step function as \((0)u(t)\), where \(u(t)\) is the unit step function.

Step 4 ：For the interval 1 <= t < 2, the function value is t - 1. This can be represented by the unit step function as \((t - 1)u(t - 1)\), where \(u(t - 1)\) is the unit step function shifted by 1.

Step 5 ：For the interval t >= 2, the function value is \(t^2\). This can be represented by the unit step function as \(t^2u(t - 2)\), where \(u(t - 2)\) is the unit step function shifted by 2.

Step 6 ：So, the piecewise function can be rewritten as \(f(t) = (0)u(t) + (t - 1)u(t - 1) - (t - 1)u(t - 2) + t^2u(t - 2)\).

Step 7 ：\(\boxed{f(t) = (0)u(t) + (t - 1)u(t - 1) - (t - 1)u(t - 2) + t^2u(t - 2)}\) is the final answer.