I have this analysis problem:
"Let $\gamma:\mathbb R \to \mathbb R^p$ be a curve which passes through the origin in $\mathbb R^p$ at a point where its velocity vector is non-zero (that is, assume $\gamma(t_0)=0$ and $\gamma'(t_0)\neq0$ at some point $t_0\in\mathbb R$). Prove that there is an interval $I$ centered at $t_0$ such that $\Vert\gamma(t)\Vert$ is decreasing for $t<t_0$ and increasing for $t>t_0$. Hint: $\Vert\gamma\Vert$ is increasing (decreasing) wherever $\Vert\gamma\Vert^2=\gamma\cdot\gamma$ is increasing (decreasing)."
I can't quite seem to get through it. It's clear that since $\Vert\gamma\Vert$ (and $\gamma\cdot\gamma$) takes on only nonnegative values that it should decrease then increase if the value is ever 0 (like it would be at $t_0$) presuming the function is relatively well-behaved. It's just ensuring that this function is that well-behaved that I'm having troubles with. Should I be making some assumptions about $\gamma$, such as that it's continuously differentiable or something?
As I wrote in one of my comments, even in the case of $\gamma$ being continuously differentiable the proof is not so straightforward, so I decided to give it here.
Assume that the curve $\gamma$ as in the question is of class $C^1$. We have $$ \frac{d}{dt}(\gamma(t) \cdot \gamma(t)) = 2 \gamma'(t) \cdot \gamma(t) $$ for all $t \in \mathbb{R}$. The problem reduces to finding the sign of $$ \gamma'(t) \cdot \gamma(t) $$ for $t$ close to $t_0$, $t \ne t_0$. Using the integral form of the mean value theorem for vector-valued functions we can write $$ \gamma(t) = (t - t_0) \Bigl( \int\limits_{0}^1 \gamma'(t_0 + s(t-t_0))\, ds \Bigr), $$ so $$ \gamma'(t) \cdot \gamma(t) = (t - t_0) \Bigl( \gamma'(t) \cdot \int\limits_{0}^1 \gamma'(t_0 + h(t-t_0))\, dh \Bigr) $$ for $t$ close to $t_0$. The inner product $$ \gamma'(t) \cdot \int\limits_{0}^1 \gamma'(t_0 + h(t-t_0))\, dh $$ at $t_0$ equals $\lVert \gamma'(t_0)\rVert^2 > 0$, therefore it follows from the $C^1$ property of $\gamma$ that it is positive for $t$ belonging to some neighborhood of $t_0$. This completes the proof.