As the image shows I have to write the piecewise function as a unit step function thing. I have tried to do this: $$1\cdot\Bigg(u(t-0)-u(t-1)\Bigg) + c\cdot \Bigg(u(t-1)\Bigg) = u(t) + (c-1)\cdot u(t-1)$$
but that doesn't get the same as the solution here. What do I do?
Your function $g$ will have a jump at $t=0$ and $t=1$.
For $t<0$, you have $g(t)=0$, for $0<t<1$ you have $g(t) = 1$, and for $t>1$ you have $g(t) = c$.
The function $f$ in the solution has: for $t<1$, $f(t) = 1$, and for $t>1$, $f(t) = c$.
The difference is what happens for $t<0$. But the original function doesn't define what happens for $t<0$, so presumably $(-\infty,0)$ is not in the domain of $f$. In other words, your function agrees with the solution function on the domain $[0,\infty)$.
I would say the problem statement is a little unclear.