Drawing thick points and little circles at coordinates

Question

I would like to draw thick black points on coordinates

(0,0) and (4,0)

and small little circles on coordinates

(0,4) and (4,4)

but I do not know how to do that. Can you help me?

\documentclass{article}
        \usepackage{amsmath}
        \usepackage{pgfplots}
        \usepackage{color}
        \usepackage{tikz}
        \begin{document}

        \begin{tikzpicture}
        [
        x=1cm, y=1cm,  scale=1.0, 
        font=\footnotesize,
        >=latex  
        ]


        \draw[ help lines] (0,0) grid (4.5,4.5);

        % x-axis
        \draw[->] (-0.2,0) -- (4.5,0) node[below] {$\varphi(a)$}; 
        %numbers on x-axis
        \draw[shift={(0,0)}, color=black] (0pt,2pt) -- (0pt,-2pt) node[below] {\footnotesize $0$};
        \draw[shift={(4,0)}, color=black] (0pt,2pt) -- (0pt,-2pt) node[below] {\footnotesize $4$};

        % y-axis 
        \draw[->] (0,-0.2) -- (0,4.5) node[right] {};%node[above left]
        % numbers on axis
        \draw[shift={(0,4)},color=black] (2pt,0pt) -- (-2pt,0pt) node[left] {\footnotesize $4$};
        \draw[shift={(0,2)},color=black] (2pt,0pt) -- (-2pt,0pt) node[left] {\footnotesize $2$};
        \draw[shift={(0,1.5)},color=black] (2pt,0pt) -- (-2pt,0pt) node[left] {\footnotesize $\varphi(e)$};

     \end{tikzpicture}

        \end{document}

Answer 1

after removing all superfluous stuff and added dots and circle on desired coordinates:

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

\begin{document}
    \begin{tikzpicture}
    [
    x=1cm, y=1cm,  scale=1.0,
    font=\footnotesize,
    >=latex
    ]
\draw[ help lines] (0,0) grid (4.5,4.5);
% x-axis
\draw[->] (-0.2,0) -- (4.5,0) node[below] {$\varphi(a)$};
%numbers on x-axis
\foreach \i in {0,...,4}
    \draw (\i,1mm) -- ++ (0pt,-2mm) node[below] {$\i$};
% y-axis
\draw[->] (0,-0.2) -- (0,4.5);
% numbers on axis
\foreach \i in {1,...,4}
    \draw   (0,\i) -- ++ (-2mm,0pt) node[left] {$\i$};
\draw   (1mm,1.5)  -- ++ (-2mm,0pt) node[left] {$\varphi(e)$};
% cirles (marks)
\fill   (0,0) circle (1mm) (4,0) circle (1mm); % <--- added
\draw[fill=white]
        (0,4) circle (1mm) (4,4) circle (1mm); % <--- added
    \end{tikzpicture}
\end{document}