Showing a regular $C^1$ function has a strong tangent

by infinitylord   Last Updated January 17, 2018 02:20 AM

Let $F: I \subset \mathbb{R} \to \mathbb{R}^3$ be a $C^1$ function which is regular at $t = t_0$. Show that $F$ has a strong tangent at $t_0$.

The definition of strong tangent I am using is that the line connecting $F(t_0 + h)$ and $F(t_0 + k)$ has a limit position as $h,k \to 0$.

To start, I noted that the line connecting $F(t_0 + h)$ and $F(t_0 + k)$ can be written as:

$$L(\lambda) = F(t_0 + k) + \lambda( F(t_0 + h) - F(t_0 + k) )$$

I thought I may try to make this look more like a derivative, so I rewrote it as:

$$L(\lambda) = F(t_0 + k) + \lambda[ (F(t_0 + h) - F(t_0)) - (F(t_0 + k) - F(t_0)) ]$$

since $F$ is $C^1$ and regular at $t_0$, $F'(t_0)$ exists and is non-zero. I then tried to write the differentiation definition somewhat differently than normal to fit the line parametrization terms:

$$F(t_0 + h) - F(t_0) = F'(t_0)h + \epsilon_1h $$ and identically, $$F(t_0 + k) - F(t_0) = F'(t_0)k + \epsilon_2k $$ where $\epsilon_1, \epsilon_2 \to (0,0,0)$ as $h,k \to 0$, respectively. (would there be any loss in generality by assuming $\epsilon_1 = \epsilon_2$?)

So that $$L(\lambda) = F(t_0 + k) + \lambda[ F'(t_0)(h + k) + \epsilon_1h + \epsilon_2k]$$

This seemed like a start to me, and at a glance appears to be telling me the line will have a limit position in the direction of $F'(t_0)$, but I'm not sure how to proceed in a more analytic sense.

I still don't feel particularly comfortable with proving limit positions for lines except in $\mathbb{R^2}$ where I can easily cite the slope. Moreover, I have seen no place where the regularity has been made useful, so I imagine it has something to do with the analytic interpretation of strong tangent I am missing here.

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