Good day! For my introductory Analysis course, I have to solve the following problem:
Let $f:\mathbb R^p \rightarrow \mathbb R$ be partially differentiable with continuous gradient in $a\in \text{Dom}(f)$ and $\text{grad}\ f(a)\ne\vec{0}$. Show that there exists a curve $c:I\rightarrow \mathbb R^p$, with $I$ some interval consisting of more than one point, such that $f(c(t))=t$ for all $t\in I$.
Now I have found this page, discussing the same problem. However, the page did not contain an answer that made sense to me.
That said, I have tried to approach the problem as follows. In another exercise (say EX1), I have proven that, for a continuously differentiable function $g:\mathbb R\rightarrow\mathbb R$ with $a\in\text{Dom}(g)$ and $g'(a)\ne0$, there exist intervals $I,J\subset\mathbb R$ which consist of more than one point, such that there exists a function $h:J\rightarrow I$ with $\forall y\in J:g(h(y))=y$ and $\forall x\in I: h(g(x))=x$. Since $\text{grad}\ f$ is nonzero in $a$, there exists a partial derivative $D_kf$, with $1\le k\le p$ such that $D_kf(a)\ne 0$. Also, since $\text{grad}\ f$ is continuous in $a$, $D_kf$ is continuous in $a$. Now notice that $D_kf$ is essentially a function from $\mathbb R^p$ to $\mathbb R$ but can also be viewed as a function from $\mathbb R$ to $\mathbb R$: it is the derivative of the function $l:\mathbb R\rightarrow \mathbb R$, $t\mapsto f(a_1,...,a_{k-1},t,a_{k+1},...,a_p)$, with $a=(a_1,...,a_p)$. We know that this function is continuously differentiable in $a_k$. Now if this function would be continuously differentiable in a neighborhood of $a_k$ (consisting of more than one point), I could use EX1 to show that there exist intervals $I,J\subset\mathbb R$ which consist of more than one point, such that there exists a function $c^*:J\rightarrow I$ with $\forall y\in J:l(c^*(y))=y$ and by defining $c:I\rightarrow \mathbb R^p$ as $c(t)=(a_1,...,a_{k-1},c^*(t),a_{k+1},...,a_p)$ one can see that $f(c(t))=t$ for all $t\in I$.
Now I have a few problems regarding the proof. To use my method, I'd have to show that $l$ as defined above should be continuously differentiable on some interval containing $a_k$, but I don't really know how to prove this or wether it is true at all. Another one of my concerns is wether that last substitution (of $c^*$) makes sense at all.
I was wondering if someone could help me solve this problem (perhaps by concidering a different approach; please keep in mind that I do not understand very advanced analysis, since I am only following an introductory course) or point out some potential errors that I made in my reasoning. Thanks!