c=10; d=1; 
L=5; e=.5; 
a=7; b=13;
f[x_]:=x/2
 Plot[f[x],{x,a,b},GridLinesStyle->{Directive[Thick,Dashed,Red],Directive[Thick,Dashed,Black]},
Epilog ->{PointSize[.02],Point[{c,L}]}]
Plot[f[x],{x,a,b},GridLines->{{},{L-e,L+e}},GridLinesStyle->{Directive[Thick,Dashed,Red],Directive[Thick,Dashed,Black]},
Epilog ->{PointSize[.02],Point[{c,L}]}]
Plot[f[x],{x,a,b},GridLines->{{c-d,c+d},{L-e,L+e}},GridLinesStyle->{Directive[Thick,Dashed,Red],Directive[Thick,Dashed,Black]},
Epilog ->{PointSize[.02],Point[{c,L}]}]

c=4; d=.1; 
L=16; e=1; 
a=3.5; b=4.5;
f[x_]:=x^2
 Plot[f[x],{x,a,b},GridLines->{{c-d,c+d},{L-e,L+e}},GridLinesStyle->{Directive[Thick,Dashed,Red],Directive[Thick,Dashed,Black]},Epilog ->{PointSize[.02],Point[{c,L}]}]

c=4; d=1.75; 
L=2; e=.5; 
a=0; b=9;
f[x_]:=Sqrt[x]
 Plot[f[x],{x,a,b},GridLines->{{c-d,c+d},{L-e,L+e}},GridLinesStyle->{Directive[Thick,Dashed,Red],Directive[Thick,Dashed,Black]},Epilog ->{PointSize[.02],Point[{c,L}]}]

c=9; d=.9; 
L=.1; e=.01; 
a=7; b=11;
f[x_]:=1/(1+x)
 Plot[f[x],{x,a,b},GridLines->{{c-d,c+d},{L-e,L+e}},GridLinesStyle->{Directive[Thick,Dashed,Red],Directive[Thick,Dashed,Black]},Epilog ->{PointSize[.02],Point[{c,L}]}]

Plot[f[x],{x,-2,2},PlotStyle->Thickness[.01],PlotRange->{-.5,1.5}]

c=0; d=0; 
L=1; e=2; 
a=-2; b=2;
f[x_]:=Piecewise[{{1,x>=0},{0,x<0}}]
 Plot[f[x],{x,a,b},PlotStyle->Thickness[.01],GridLines->{{c-d,c+d},{L-e,L+e}},GridLinesStyle->{Directive[Thick,Dashed,Red],Directive[Thick,Dashed,Black]},Epilog ->{PointSize[.02],Point[{c,L}]},PlotRange->{-2,3}]

c=0; d=Sqrt[1/4]; 
L=0; e=1/4; 
a=-1; b=1;
f[x_]:=x^2
 Plot[f[x],{x,a,b},PlotStyle->Thickness[.01],GridLines->{{c-d,c+d},{L-e,L+e}},GridLinesStyle->{Directive[Thick,Dashed,Red],Directive[Thick,Dashed,Black]},Epilog ->{PointSize[.02],Point[{c,L}]},PlotRange->{-1,1}]

c=0; d=Sqrt[1/4]; 
L=0; e=1/4; 
a=-1; b=1;
f[x_]:=x^2
 Plot[f[x],{x,a,b},PlotStyle->Thickness[.01],GridLines->{{c-d,c+d},{L-e,L+e}},GridLinesStyle->{Directive[Thick,Dashed,Red],Directive[Thick,Dashed,Black]},Epilog ->{PointSize[.02],Point[{c,L}]},PlotRange->{-1,

NumberLinePlot[{Interval[{1,9}],Interval[{1,7}],4}]

Plot[Sin[1/x],{x,-1/2,1/2},Axes->Thick,Ticks->{{},{}}]

d[x_]:=Piecewise[{{1,Abs[Rationalize[x,0.01]-x]>0.004},{0,True}}];
DiscretePlot[d[x],{x,0,1,.005},Ticks->{{},{}}]

maxq=100;
fracs=Table[p/q,{q,2,maxq},{p,2,q}]//Flatten//DeleteDuplicates;
pq={#,1/Denominator@#}&/@fracs;

ListPlot[pq,PlotRange->{0,.5},AxesStyle->Directive[16,Thick,Black],Ticks->{{},{}}]

Plot[x Sin[1/x],{x,-1/2,1/2},Axes->Thick,Ticks->{{},{}}]

Plot[{x Sin[1/x],x,-x},{x,-1/2,1/2},PlotStyle->{{Thick},{Thick,Dashed},{Thick,Dashed}},Axes->Thick,Ticks->{{},{}}]

Plot[{x Sin[1/x],Abs[x],-Abs[x]},{x,-1/2,1/2},PlotStyle->{{Thick},{Thick,Dashed},{Thick,Dashed}},Axes->Thick,Ticks->{{},{}}]

Plot[{Abs[x Sin[1/x]],Abs[x],0},{x,-1/2,1/2},PlotStyle->{{Thick},{Thick,Dashed},{Thick,Dashed}},Axes->Thick,Ticks->{{},{}}]

Plot[{(x-3)(5Sin[1/(x-3)] - 2 ),Abs[3(x-3)],-Abs[7(x-3)]},{x,2.5,3.5},PlotStyle->{{Thick},{Thick,Dashed},{Thick,Dashed}},Axes->Thick,Ticks->{{},{}}]

Plot[{Abs[(x-3)(5Sin[1/(x-3)] - 2 )],Abs[3(x-3)]},{x,2.5,3.5},PlotStyle->{Thick,{Thick,Dashed},{Thick,Dashed}},Axes->Thick,Ticks->{{},{}}]