If the composition of a function with a curve gives you no local minima or maxima do I have a saddle point?

Question

If I have a function $f$ with a critic point $P$ and a continuous curve $\alpha$ such that $\alpha(t_o)=P $ and $(f\circ\alpha)$has no local maxima nor local minima in $t_o$. Can I affirm that $P$ is a saddle point?

Answer 1

Let an arbitrary neighborhood $V$ of $P$ be given. Since $\alpha$ is continuous at $t_0$ there is a neighborhood $U$ of $t_0$ with $\alpha(U)\subset V$.

Since the function $\psi:=f\circ \alpha$ has no local maximum at $t_0$ the neighborhood $U$ contains a point $t_>$ with $\psi(t_>)>\psi(t_0)$. Similarly, since $\psi$ has no local minimum at $t_0$ the neighborhood $U$ contains a point $t_<$ with $\psi(t_<)<\psi (t_0)=f(P)$.

Now the points $P_>:=\alpha(t_>)$ and $P_<:=\alpha(t_<)$ are lying in $V$, and $$f(P_>)=\psi(t_>)>\psi(t_0)=f(P),\quad f(P_<)=\psi(t_<)<\psi(t_0)=f(P)\ .$$ Since $V$ was arbitrary this shows that $f$ has neither a local maximum nor a local minimum at $P$. Since $P$ is a critical point of $f$ we therefore have a saddle point.