How to customize tick text in TikZ?

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A suggestion with pgfplots:

I think, you want something like that...

· One methode is setting extra x ticks, see "T_0+T_1"; same game with "K" by using extra y ticks.

· Another method is to add special labels as a draw annotation, see "T_0", "T_1" and "T_a", "T_b":

If you want to use different tick lenghts for your labels, you could prefer the second method.

The following shows both ways.

\documentclass[border=5pt, tikz]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=newest}
\usetikzlibrary{backgrounds}
\begin{document}
\pgfdeclarelayer{foreground}
\pgfsetlayers{background,main,foreground}
\begin{tikzpicture}
\pgfmathsetlengthmacro\MajorTickLength{
\pgfkeysvalueof{/pgfplots/major tick length}*0.5}
\pgfmathsetlengthmacro\AxisShift{-4.5pt}
\begin{axis}[ 
xmin=-0, 
xmax=6.0, 
ymin=0,
ymax=4.5,
%axis lines=middle,
axis y line=middle, 
axis x line=bottom,
axis line style={-, gray},
major tick length=\MajorTickLength,
every tick/.style={gray},
tick align=outside,
y axis line style={xshift=\AxisShift},
every y tick/.style={xshift=\AxisShift},
yticklabel style={xshift=\AxisShift},
x axis line style={yshift=\AxisShift},
every x tick/.style={yshift=\AxisShift},
x tick label style={yshift=\AxisShift, fill=white},
ylabel={\rotatebox{90}{$y(t)$}},
xlabel={$t$},
x label style={at={(ticklabel* cs:0.975)}, inner sep=5pt, yshift=4*\AxisShift, anchor=north},
y label style={at={(ticklabel* cs:0.55)}, inner sep=5pt, xshift=3.5*\AxisShift, anchor=east},
xtick={0,...,6},
ytick={1,...,4},
extra y ticks={0,4.5},
extra y tick labels={0,$K$},
grid=major, 
extra x ticks={2},
extra x tick labels={$T_0 + T_1$},
extra x tick style={major tick length=-5.25*\AxisShift, on background layer},
%enlarge y limits={abs=0.4,upper},
]
% Curve
\addplot [domain=1.5:6, samples=222, thick]{9/2 *(1 -exp(3 -2*x)};
% Line 1 as parametric plot
\addplot[dashed, variable=\t, samples=2, thick]({2},{\t}) ;       
% Line 2 as table-plot
\addplot[no marks, thick,] table[x=X,y=Y] {
X     Y
1.5   0
2     4.5
};
% Some special labels ===================
\begin{pgfonlayer}{foreground}
\foreach \T/\Name in {0.5/T_1, 1.5/T_0, 3.85/T_a, 4.6/T_b} {\edef\temp{\noexpand
\draw [gray, yshift=\AxisShift] (\T,0) -- (\T,2.75*\AxisShift) node[below, text=black, inner sep=2pt]{$\Name$}; 
}\temp}
\end{pgfonlayer}
% ================================
\end{axis}   

\end{tikzpicture}
\end{document}   

This does something of that sort. In order for one additional tick not wiping out the previous one, you need to the /.list key. In order to avoid overlaps, you can use shifts. So altogether

major also at/.list={ 1.5 as    $T_0$,
           2 as [node style={yshift=-2em}] $T_0 + T_1$},

Full MWE:

\documentclass{scrartcl}
\usepackage{tikz}
\usetikzlibrary{datavisualization.formats.functions}

\begin{document}

  \begin{tikzpicture}
    \datavisualization[
       scientific axes={clean},
       all axes = grid,
       x axis = {
          min value = 0,
          ticks = {
               major also at/.list={ 1.5 as $T_0$,
               2 as [node style={yshift=-2em,alias=pft}] $T_0 + T_1$},
             stack
            },
          label = $t$
         },
       y axis = {
          min value = 0,
          ticks = {major = {also at = 4.5 as $K$}},
          label = $y(t)$
         },
       visualize as smooth line/.list = {
          curve,
          line 1,
          line 2
         },
       line 2 = {style = dashed}
      ]
    data[
         set = curve,
         format = function
        ] {
           var x : interval[1.5 : 6];
           func y = 9/2 * (1 - exp(3 - 2 * \value x));
          }
    data[set = line 1] {
                        x,   y
                        1.5, 0
                        2,   4.5
                       }
    data[set = line 2] {
                        x, y
                        2, 4.5
                        2, 0
                       };
   \draw[very thin] (pft.north) -- ++ (0,1em);
  \end{tikzpicture}

\end{document}