# How to express this convolution by the sum of integrals

If $$f\left(x\right)=\begin{cases} f_{1}\left(x\right), & x\in[0,1]\\ f_{2}\left(x\right), & x\in[1,\sqrt{5}]\\ 0, & \mbox{elsewhere} \end{cases}$$ what does the piecewise-defined function $f\star f$ look like or how to express the pieces of $f\star f$ by the integrals (or the sum of integrals) of the products of $f_{1}$ and $f_{2}$? Thanks in advance.

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If $$f\left(x\right)=\begin{cases} f_{1}\left(x\right), & x\in[0,a_1)\\ f_{2}\left(x\right), & x\in[a_1,a_2)\\ f_3(x), & x\geq a_2 \end{cases}$$ then by using the unit step function, we can write $f(x)$ as the following summation: $$f(x)=f_1(x)+\left(f_2(x)-f_1(x)\right)u_{a_1}(x)+\left(f_3(x)-f_2(x)\right)u_{a_2}(x)$$ wherein $$u_c(x)=\begin{cases} 0\left(x\right), & x\in[0,c)\\ 1\left(x\right), & x\geq c\\ \end{cases}$$ is step unit function.