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How to scale down the size of tick (j-1, j and j+1) of a graph in Tikz?

Moreover, as I don't have a lot of experience with Tikz, if there is a more standard/efficient way of writing the whole graph, I'm obviously glad to learn!

I obtained this graph by modifying a an example that I found on Texample.net

documentclass[a4paper, 12pt]{book}

usepackage{pgfplots, tikz}

begin{document}

begin{tikzpicture}[scale = 0.7]

% Draw axes

draw [<->,thick] (0,5) node (yaxis) [above] {$y$}

|- (5,0) node (xaxis) [right] {$x$};



coordinate (c) at (2,2);

fill[red] (c) circle (3pt);

draw[dashed] (yaxis |- c) node[left] {$i$}

-| (xaxis -| c) node[below] {$j-1$};



coordinate (d) at (3,2);

fill[red] (d) circle (3pt);

draw[dashed] (yaxis |- d) node[left] {$i$}

-| (xaxis -| d) node[below] {$j$};



coordinate (d) at (3,3);

fill[red] (d) circle (3pt);

draw[dashed] (yaxis |- d) node[left] {$i+1$}

-| (xaxis -| d) node[below] {$j$};



coordinate (d) at (4,2);

fill[red] (d) circle (3pt);

draw[dashed] (yaxis |- d) node[left] {$i$}

-| (xaxis -| d) node[below] {$j+1$};



draw [thin, gray] (0,0) grid (5,5);

end{tikzpicture}

end{document}

I put some opacity in the grid. In my opinion, they should also put the red dots in the foreground to hide the dotted lines that invade the dots. But I don't know if you want this.

documentclass[a4paper, 12pt]{book}

usepackage{pgfplots, tikz}

begin{document}

begin{tikzpicture}[scale = 0.7]

% Draw axes

draw [<->,thick] (0,5) node (yaxis) [above] {$y$}

|- (5,0) node (xaxis) [right] {$x$};



coordinate (c) at (2,2);

fill[red] (c) circle (3pt);

draw[dashed] (yaxis |- c) node[left] {}

-| (xaxis -| c) node[below] {$scriptstyle{j-1}$};



coordinate (d) at (3,2);

fill[red] (d) circle (3pt);

draw[dashed] (yaxis |- d) node[left] {$scriptstyle{i}$}

-| (xaxis -| d) node[below] {};



coordinate (d) at (3,3);

fill[red] (d) circle (3pt);

draw[dashed] (yaxis |- d) node[left] {$scriptstyle{i+1}$}

-| (xaxis -| d) node[below] {$scriptstyle{j}$};



coordinate (d) at (4,2);

fill[red] (d) circle (3pt);

draw[dashed] (yaxis |- d) node[left] {}

-| (xaxis -| d) node[below] {$scriptstyle{j+1}$};



draw [thin, gray] (0,0) grid (5,5);

end{tikzpicture}

end{document}

If you don't mind that x and y will also become smaller, you could just use a smaller font size, e.g. by using scriptsize before the tikzpicture. This will also avoid the problem, that scaling elements that contain text will result in a suboptimal choice of letter shapes (see Why not scale elements that contain text for more information).

Another small problem: you print the labels for i and j three times above each other, depending on the rendering precision of your pdf viewer this can cause the font to look awkward, better only print it one time.

It would also be better, if you would choose unique names for your coordinates instead of re-using d for 3 of them. This has the big advantage, that you can easily control the order in which you draw things. For example you can first draw the grid in the background, then the axis, the dashed lines and the red dots at the end.

documentclass[a4paper, 12pt]{book}

usepackage{pgfplots, tikz}

begin{document}



begingroup

    scriptsize

    begin{tikzpicture}[scale=0.7]



        % grid

        draw [thin, gray] (0,0) grid (5,5);



        % Draw axes

        draw [<->,thick] (0,5) node (yaxis) [above] {$y$} |- (5,0) node (xaxis) [right] {$x$};



        % Coordinates   

        coordinate (a) at (2,2);

        coordinate (b) at (3,2);

        coordinate (c) at (3,3);

        coordinate (d) at (4,2);



        % Dashed lines

        draw[dashed] (yaxis |- a) node[left] {} -| (xaxis -| a) node[below] {$j-1$};

        draw[dashed] (yaxis |- b) node[left] {} -| (xaxis -| b) node[below] {};

        draw[dashed] (yaxis |- c) node[left] {$i+1$} -| (xaxis -| c) node[below] {$j$};

        draw[dashed] (yaxis |- d) node[left] {$i$} -| (xaxis -| d) node[below] {$j+1$};



        % red dots

        fill[red] (a) circle (3pt);

        fill[red] (b) circle (3pt);

        fill[red] (c) circle (3pt);

        fill[red] (d) circle (3pt);



    end{tikzpicture}

endgroup

end{document}