As funções de Heaviside e rampa

\[\begin{array}{*{20}{c}}
{H\left( x \right) = \left\{ {\begin{array}{*{20}{c}}
0& \Leftarrow &{x < 0} \\
{\frac{1}{2}}& \Leftarrow &{x = 0} \\
1& \Leftarrow &{x > 0}
\end{array}} \right.}&{\text{e}}&{R\left( x \right) = \left\{ {\begin{array}{*{20}{c}}
0& \Leftarrow &{x \leqslant 0} \\
x& \Leftarrow &{x > 0}
\end{array}} \right.}
\end{array}\]

1. \[\begin{array}{*{20}{c}}
   {x\,H\left( x \right)}& = &{x \times \left\{ {\begin{array}{*{20}{c}}
   0& \Leftarrow &{x < 0} \\
   {\frac{1}{2}}& \Leftarrow &{x = 0} \\
   1& \Leftarrow &{x > 0}
   \end{array}} \right.}& = &{\left\{ {\begin{array}{*{20}{c}}
   {x \times 0}& \Leftarrow &{x < 0} \\
   {x \times \frac{1}{2}}& \Leftarrow &{x = 0} \\
   {x \times 1}& \Leftarrow &{x > 0}
   \end{array}} \right.}& = &{\left\{ {\begin{array}{*{20}{c}}
   0& \Leftarrow &{x < 0} \\
   0& \Leftarrow &{x = 0} \\
   x& \Leftarrow &{x > 0}
   \end{array}} \right.}& = &{R\left( x \right)}
   \end{array}\]
2. \[\begin{array}{*{20}{c}}
   {\frac{{x + \left| x \right|}}{2}}& = &{\left\{ {\begin{array}{*{20}{c}}
   {\frac{{x + \left( { – x} \right)}}{2}}& \Leftarrow &{x \leqslant 0} \\
   {\frac{{x + x}}{2}}& \Leftarrow &{x > 0}
   \end{array}} \right.}& = &{\left\{ {\begin{array}{*{20}{c}}
   0& \Leftarrow &{x \leqslant 0} \\
   x& \Leftarrow &{x > 0}
   \end{array}} \right.}& = &{R\left( x \right)}
   \end{array}\]
3. \[\begin{array}{*{20}{c}}
   {\left( {R \circ R} \right)\left( x \right)}& = &{R\left( {R\left( x \right)} \right)}& = &{R\left( {\left\{ {\begin{array}{*{20}{c}}
   0& \Leftarrow &{x \leqslant 0} \\
   x& \Leftarrow &{x > 0}
   \end{array}} \right.} \right)}& = &{R\left( x \right)}
   \end{array}\]