I have a function $f:\mathbb{R} \to \mathbb{R}$ with $f(0)=0$ and I have a sphere $S_r^1((0,r))$ in $\mathbb{R}^2$ with $r>0$ given by the equation $x^2+(y-r)^2=r^2$ which touches the point of origins with its lowest point.
I know that if $f$ is $C^2$ you can rotate $f$ in such a way that $|f|$ won't intersect the sphere in any point but the origin (because for a $C^2$-function you can assume $f(0)=\nabla f(0)=0$ after translation and rotation, and with Taylor's formula (which uses the $C^2$-property) $f$ is majorized by a parabola in which you always can put a sphere).
That doesn't work with e.g. the absolute value $g(x)=|x|$ which is only $C^0$, because you will intersect the sphere at the points $x=\pm r$.
My question is, is there a $C^1$-function $f$ which is not $C^2$ with $f(0)=0$ where $|f|$ intersects the sphere only in the origin?
I tried to use $|x|^s$ with $1<s<2$, but for small $x \ll r$ it still intersects the sphere.
How about $$f(x) = \cases{0 & if $x \le 0$\cr x^2 & if $x > 0$\cr} $$