Vrftjkry

This Cusp I made came out awkward and I can not fix it.

documentclass{article}

usepackage{tikz}

begin{document}

begin{tikzpicture}[scale=1,cap=round]

 tikzset{axes/.style={}}

 % The graphic

begin{scope}[style=axes]

draw[->] (-.5,0) -- (3,0) node[below] {$x$};

draw[->] (0,-.5)-- (0,3) node[left] {$y$};

draw (0.25,0.4) to [out=10,in=80] (1.5,2.5);

draw (1.5,2.5) to [out=-80, in=175] (2.75,.4);

%%%

end{scope}

end{tikzpicture}

end{document}

This outputs:

I am trying to achieve:

Note it is not exactly symmetric either but the head is cleaner.

I guess the sign of 80 in draw (0.25,0.4) to [out=10,in=80] (1.5,2.5); is the culprit.

documentclass{article}

usepackage{tikz}

begin{document}

begin{tikzpicture}[scale=1,cap=round]

 tikzset{axes/.style={}}

 % The graphic

begin{scope}[style=axes]

draw[->] (-.5,0) -- (3,0) node[below] {$x$};

draw[->] (0,-.5)-- (0,3) node[left] {$y$};

draw[thick] (0.25,0.4) to [out=10,in=-90]  (1.5,2.5)

 to [out=-90, in=175] (2.75,.4);

draw[thin] (1.5,2.5) -- ++ (0.2,0.6) node[above]{$f'(a)$ does not exist};

draw (1.5,0.2) -- (1.5,-0.2) node[below]{$a$};

%%%

end{scope}

end{tikzpicture}

end{document}

Here is a solution using the pgfplots package, which extends tikz to include a wide variety of plotting options:

documentclass{article}

usepackage{pgfplots}

begin{document}

begin{tikzpicture}begin{axis}[xmin=0,xmax=1,ymin=0,xtick={0.5},xticklabels=$a$,axis lines=left,ymajorticks=false,xlabel=$x$,ylabel=$y$,clip=false]

addplot  [domain=0.1:0.5] {0.1+x^2} node [pos=0.2,above left] {$f$};

addplot  [domain=0.5:0.9] {0.1+(x-1)^2} node[pin={90: $f'(a)$ does not exist.} ] at (axis cs:0.5,0.35) {};

end{axis}

end{tikzpicture}

end{document}

A PSTricks solution just for comparison purposes.

documentclass[pstricks]{standalone}

usepackage{pst-plot,pst-plot}

deff#1{(x-#1)^2+1}

begin{document}

begin{pspicture}[algebraic,ticks=none,labels=none](-.4,-.5)(4.5,3.5)

psaxes{->}(0,0)(-.3,-.4)(4,3)[$x$,0][$y$,90]

psplot{1}{2}{f{1}}

psplot{2}{3}{f{3}}

psxTick[labelsep=1pt](2){a}

rput(2.2,3){rnode[b]{a}{$f'(a)$ does not exist}}

pcline[nodesep=2pt]{->}(a)(*2 {f{1}})

end{pspicture} 

end{document}