SetOptions[Plot,AspectRatio->Automatic];
tf:=TableForm;tx:=TeXForm;hh:=Hold;
ss:=Simplify;
tt[x_]:=TraditionalForm[x];
mf:=MatrixForm;


split[x_]:=2 x/;x<.5;split[x_]:=3(x-1)^2/;x>.5

Split

Plot[split[x],{x,0,1},AspectRatio->1,Exclusions->{.5}]

Finite

finite[1]=3;finite[2]=3;finite[3]=2;finite[6]=1

nodel[ch_,p_]:=Text[Style[ch,14,Bold],p]

Module[{s=.18,a=.15},Show[
Graphics[{Arrowheads[a],
nodel["a",{1,4}],nodel["b",{1,3}],nodel["c",{1,2}],nodel["d",{1,1}],nodel["a",{0,4}],nodel["b",{0,3}],nodel["c",{0,2}],nodel["d",{0,1}],Arrow[{{0,4},{1,4}},s],Arrow[{{0,3},{1,2}},s],Arrow[{{0,2},{1,2}},s],Arrow[{{0,1},{1,3}},s]
}
],
ImageSize->Tiny,AspectRatio->1.5]
]

Fresnel

fs[x_]:=Integrate[Sin[t^2],{t,0,x}]

fs[2]

fs[2]//N

units[xmin_,xmax_]:=Range[Floor[xmin],Floor[xmax],1]

fsplot=Plot[fs[x],{x,0,5},PlotRange->{{0,5},{-1,1}},AspectRatio->.4,Ticks->units]

Sine Blur

sb[x_]:=Sin[10/x]

Plot[sb[x],{x,0,2},AspectRatio->.5]

Derivative

st2[x_]:=Sin[x^2]

st2plot=Plot[st2[x],{x,0,5},AspectRatio->.4, Ticks->units]

Show[
Graphics[
{
Inset[fsplot,{0,0}],
Inset[Style[" ⟹ ",16],{1.6,0}],
Inset[st2plot,{3,0}]
},AspectRatio->.3
]
]