a=(1/2);
bb=15;
f[x_]=(∑_(n=0)^25)((a^n)Cos[(bb^n)π x]);

Manipulate[Plot[f[x],{x,-3(1-t)+0t,3(1-t)+0t},PlotRange->{-2(1-t)+2t,2},PlotPoints->200,Ticks->None],

(*Manipulate Controllers*)
Style["The Weierstrass Function",Bold,12],(*Title*)
Style["Continuous everywhere and ",Plain,12],
Style["differentiable nowhere",Plain,12],
Delimiter,

{{t,0,"zoom"},0,0.999,ImageSize->Tiny},ControlPlacement->Left]

Manipulate[
With[{f=If[x<=0.5,x/2,k(x-0.5)+b]},Show[Plot[f,{x,.1,.7},Ticks->None,PlotStyle->{Thick},Exclusions->{0.5},PlotRange->{{-.2,1},{-.2,1}}],Graphics[{{Point[{0,0.05}],Point[{0,k(.7-0.5)+b}],Point[{0.1,0}],Point[{.7,0}]},{RGBColor[.49, 0, 0],PointSize[0.02],Point[{0,y}]},{If[0.05≤y≤0.25 || b<y<=k(.7-0.5)+b,{RGBColor[.33, .6, .7],Line[{{0,y},{If[0.05≤y≤0.25 ,2y,(y-b)/k+.5],y}}],Arrow[{{If[0.05≤y≤0.25 ,2y,(y-b)/k+.5],y},{If[0.05≤y≤0.25 ,2y,(y-b)/k+.5],0}}],Text[Style["c",16,Italic],{If[0.05≤y≤0.25 ,2y,(y-b)/k+.5],-.03}]},Text[Style[Row[{Style["c",Italic]," doesn't exist"}],16],{.2,y}]]},{RGBColor[.49, 0, 0],Text[Style["N",16,Italic],{-.03,y}]},If[hint,{Text[Style["a",16,Italic],{0.1,-0.1}],Text[Style["b",16,Italic],{0.7,-0.1}],Text[Style[Row[{Style["f",Italic],"(",Style["a",Italic],")"}],16],{-0.08,.05}],Text[Style[Row[{Style["f",Italic],"(",Style["b",Italic],")"}],16],{-0.08,k(.7-0.5)+b}]},{}]}],ImageSize->{350,350}]],

(*Manipulate Controllers*)
Style["The Intermediate Value Theorem",Bold,12],(*Title*)
Delimiter,
{{k,2,"slope"},0.5,2,.01,Appearance->"Labeled",ImageSize->Tiny},
{{b,.25,"intercept"},.25,.5,.01,Appearance->"Labeled",ImageSize->Tiny},
{{y,0.4,"move N"},.05,k(.7-0.5)+b,.01,ImageSize->Tiny,Appearance->"Labeled"},
{{hint,True,"show interval labels"},{False,True}},
SaveDefinitions->True,ControlPlacement->Left]