Manipulate[y=c^3-8c+1;y2=(c-δ)^3-8(c-δ)+1;y3=(c+δ)^3-8(c+δ)+1;good=Apply[And,Table[y-ε<z^3-8z+1<y+ε,{z,c-δ,c+δ,δ/5}]];
Row[{
Show[Plot[z^3-8z+1,{z,-3,4},PlotStyle->AbsoluteThickness[3],AspectRatio->1,ImageSize->400],
Plot[z^3-8z+1,{z,c-δ,c+δ},PlotStyle->{Red,AbsoluteThickness[5]}],
Graphics[
{
{RGBColor[0.0,0.4,0.8,0.3],Polygon[{{-3,y-ε},{4,y-ε},{4,y+ε},{-3,y+ε}}]},
{PointSize[0.03],Point[{c,y}]},
{PointSize[0.015],Point[{c-δ,y2}]},
{PointSize[0.015],Point[{c+δ,y3}]},
{AbsoluteThickness[2],
Line[{{c,0.7},{c,-0.7}}],
Line[{{c-δ,0.7},{c-δ,-0.7}}],
Line[{{c+δ,0.7},{c+δ,-0.7}}],
Line[{{-0.12,y},{0.12,y}}],
Line[{{-0.12,y+ε},{0.12,y+ε}}],
Line[{{-0.12,y-ε},{0.12,y-ε}}],
},
Text[Style[x^3-8x+1,20],{-2.1,32}],
Text[Style["c",16],{c,-2}],
{If[δ>0.25,Black,If[δ>0.16,GrayLevel[0.8],RGBColor[0,0,0,0]]],Text[Style["c-δ",13],{c-δ,-4}]},
{If[δ>0.25,Black,If[δ>0.16,GrayLevel[0.8],RGBColor[0,0,0,0]]],Text[Style["c+δ",13],{c+δ,-4}]},
Text[Style["f(c)",16],{-0.4,y}],
{If[ε>1.5,Black,If[ε>0.75,GrayLevel[0.8],RGBColor[0,0,0,0]]],Text[Style["f(c)-ε",16],{-0.6,y-ε}]},
{If[ε>1.5,Black,If[ε>0.75,GrayLevel[0.8],RGBColor[0,0,0,0]]],Text[Style["f(c)+ε",16],{-0.6,y+ε}]}
}
]
],"  ",Show[Graphics[{AbsoluteThickness[4],If[!Face,{White,Circle[]},
If[good,{Circle[],Disk[{-0.4,0.4},0.12],Disk[{0.4,0.4},0.12],Circle[{0,0},0.5,{-0.2Pi,-Pi+0.2Pi}]},
{Circle[],Disk[{-0.4,0.4},0.12],Disk[{0.4,0.4},0.12],Circle[{0,-0.75},0.5,{0.2Pi,Pi-0.2Pi}]}]]}],ImageSize->100]}],{{c,3.5},-3,4},{ε,0.05,4},{δ,0.01,0.75},{Face,{False,True}}]