Plotting a function discontinuous at one point

Without having to manually look:

 f[x_] := If[x != 1, 1, 0]
 g[x_] := Floor[x]/2
 Plot[{f[x], g[x]}, {x, 0, 5}, ExclusionsStyle -> {Dotted, Directive[Black, AbsolutePointSize[5]]}]

enter image description here

Comment

Here is a workaround,

f[x_] := If[x != 1, 1, 0]

fdata = Table[{x, f[x]}, {x, 0, 2, 0.001}];

plot = ListPlot[fdata, Joined -> True]

enter image description here

Use Exclusion as plot option:

Plot[1, {x, 0, 2}, Exclusions -> {1},ExclusionsStyle -> {PointSize -> 0.1,Red}, MaxRecursion -> 4,PlotPoints -> 100]

enter image description here

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