I'm reading Elements of Geometry for Manfredo Do Carmo and I'm stucked in this problem. The book define the strong tangent: $f$ has a strong tangent at $t$ if the line determined by $f(t+h), f(t-k)$ has a limit position when $h,k\to 0$
I'm tried to show that $f$ has a unique tangent at $t$, in this case the strong tangent is the same as the weak tangent.
I think if f has two tangents then the continuity of f′ is broken...but I don't know how show that...
If we consider the curve $f(t)=(t^3,t^2)$, $f$ doesn't have strong tangent at $t=0$ because we can aproach the tangent $L=\{(\alpha,0):\alpha\in\mathbb{R}\}$ by the lines determined by the points $(h^3,h^2),(-h^3,h^2)$
But that lines aren't tangent lines and the line $L$ is impossible to aproach by tangent lines.
Then the regularity is crucial, I don't understand how it that works in the general case.