Defining a Piecewise Function for PGFplots

Question

I want to define a piecewise function q(x), and attempted to adapt the solution to this question on using pgfmathdeclarefunction to create a unit pulse function, and this works fine. However, when I attempt to plot q(x+4)+0.5, the resulting graph is not what I would expect. However, applying the same transformation on the unit pulse function from the above mentioned link works fine.

So, is there a better way to define a piecewise defined function?

The MWE below produces the following result.

Note that the graphs on the left are as one would expect for both p(x) and p(x+4)+0.5. The graphs on the right are correct for q(x), but but incorrect for q(x+4)+0.5.

\documentclass{article}
\usepackage{amsmath}
\usepackage{pgfplots}

\newcommand{\pLabel}{
$p(x)=
\begin{cases}
    1 & 0 < x < 1\\
    0 & \text{otherwise}
\end{cases}$
}
\newcommand{\qLabel}{
$q(x)=
\begin{cases}
    x & 0 < x < 1\\
    0 & \text{otherwise}
\end{cases}$
}

\newcommand{\pShiftedLabel}{$p(x+4)+0.5$}
\newcommand{\qShiftedLabel}{$q(x+4)+0.5$}

\pgfmathdeclarefunction{p}{1}{%
  \pgfmathparse{(and(#1>0, #1<1))}%
}

\pgfmathdeclarefunction{q}{1}{%
  \pgfmathparse{(and(#1>0, #1<1)*x)}%
}

\tikzstyle{MyStyle}=[domain=-5:5, samples=50, ultra thick]
\tikzstyle{pLabelStyle}=[above, yshift=22ex, xshift=-10ex]
\tikzstyle{qLabelStyle}=[below, yshift=-2ex, xshift=-10ex]
\tikzstyle{ShiftedLabelStyle}=[above left, xshift=1ex]


\begin{document}
%------------------ Using \pgfmathdeclarefunction -----------
Plot of $p(x)$ and \pShiftedLabel using PGF Version \pgfversion, followed by a plot of $q(x)$ and \qShiftedLabel

\begin{tikzpicture}
  \begin{axis}
    \addplot[MyStyle, blue]{p(x)} node [pLabelStyle] {\pLabel};
    \addplot[MyStyle, red]{p(x+4)+0.5} node [ShiftedLabelStyle] {\pShiftedLabel};
  \end{axis}
\end{tikzpicture}
%
\begin{tikzpicture}
  \begin{axis}
    \addplot[MyStyle, blue]{q(x)} node [qLabelStyle] {\qLabel};
    \addplot[MyStyle, red]{q(x+4)+0.5} node [ShiftedLabelStyle] {\qShiftedLabel};
  \end{axis}
\end{tikzpicture}

% --------------------- Using "declare function" -------------
Using declare function to define localp(x) and localq(x):

\begin{tikzpicture}
[declare function={localp(\t) =  and(\t > 0, \t < 1);}]
  \begin{axis}
    \addplot[MyStyle, blue]{localp(x)}  node [pLabelStyle] {\pLabel};
    \addplot[MyStyle, red]{localp(x+4)+0.5} node [ShiftedLabelStyle] {\pShiftedLabel};
  \end{axis}
\end{tikzpicture}
%
\begin{tikzpicture}
[declare function={localq(\t) = (and(\t > 0, \t < 1)*x);}]
  \begin{axis}
    \addplot[MyStyle, blue]{localq(x)} node [qLabelStyle] {\qLabel};
    \addplot[MyStyle, red]{localq(x+4)+0.5} node [ShiftedLabelStyle] {\qShiftedLabel};
  \end{axis}
\end{tikzpicture}
\end{document}

Answer 1

Both methods for defining piecewise functions are fine, but you should use

\pgfmathdeclarefunction{q}{1}{%
  \pgfmathparse{(and(#1>0, #1<1)*#1)}%
}

instead of and(#1>0, #1<1)*x), and

[declare function={localq(\t) = (and(\t > 0, \t < 1)*\t);}]

instead of [declare function={localq(\t) = (and(\t > 0, \t < 1)*x);}], because you don't actually want the function value to be x, but rather the value of the argument (x+4 in this case).

\documentclass{article}
\usepackage{amsmath}
\usepackage{pgfplots}

\newcommand{\pLabel}{
$p(x)=
\begin{cases}
    1 & 0 < x < 1\\
    0 & \text{otherwise}
\end{cases}$
}
\newcommand{\qLabel}{
$q(x)=
\begin{cases}
    x & 0 < x < 1\\
    0 & \text{otherwise}
\end{cases}$
}

\newcommand{\pShiftedLabel}{$p(x+4)+0.5$}
\newcommand{\qShiftedLabel}{$q(x+4)+0.5$}

\pgfmathdeclarefunction{p}{1}{%
  \pgfmathparse{(and(#1>0, #1<1))}%
}

\pgfmathdeclarefunction{q}{1}{%
  \pgfmathparse{(and(#1>0, #1<1)*#1)}%
}

\tikzstyle{MyStyle}=[domain=-5:5, samples=100, ultra thick]
\tikzstyle{pLabelStyle}=[above, yshift=22ex, xshift=-10ex]
\tikzstyle{qLabelStyle}=[below, yshift=-2ex, xshift=-10ex]
\tikzstyle{ShiftedLabelStyle}=[above left, xshift=1ex]


\begin{document}
%------------------ Using \pgfmathdeclarefunction -----------
Plot of $p(x)$ and \pShiftedLabel using PGF Version \pgfversion, followed by a plot of $q(x)$ and \qShiftedLabel

\begin{tikzpicture}
  \begin{axis}
    \addplot[MyStyle, blue]{p(x)} node [pLabelStyle] {\pLabel};
    \addplot[MyStyle, red]{p(x+4)+0.5} node [ShiftedLabelStyle] {\pShiftedLabel};
  \end{axis}
\end{tikzpicture}
%
\begin{tikzpicture}
  \begin{axis}
    \addplot[MyStyle, blue]{q(x)} node [qLabelStyle] {\qLabel};
    \addplot[MyStyle, red]{q(x+4)+0.5} node [ShiftedLabelStyle] {\qShiftedLabel};
  \end{axis}
\end{tikzpicture}

% --------------------- Using "declare function" -------------
Using declare function to define localp(x) and localq(x):

\begin{tikzpicture}
[declare function={localp(\t) =  and(\t > 0, \t < 1);}]
  \begin{axis}
    \addplot[MyStyle, blue]{localp(x)}  node [pLabelStyle] {\pLabel};
    \addplot[MyStyle, red]{localp(x+4)+0.5} node [ShiftedLabelStyle] {\pShiftedLabel};
  \end{axis}
\end{tikzpicture}
%
\begin{tikzpicture}
[declare function={localq(\t) = (and(\t > 0, \t < 1)*\t);}]
  \begin{axis}
    \addplot[MyStyle, blue]{localq(x)} node [qLabelStyle] {\qLabel};
    \addplot[MyStyle, red]{localq(x+4)+0.5} node [ShiftedLabelStyle] {\qShiftedLabel};
  \end{axis}
\end{tikzpicture}
\end{document}