Cfrgtkky

I have the following exercise I`m struggling with :

Let $fin C^1(mathbb{R}^n,mathbb{R}^m)$, and $a in mathbb{R}^n$, I need to show that $rk(D_f(x))geq rk(D_f(a))$ for some neighbourhood of $a$.

I first tried looking at non-trivial linear combinations of $nabla f_i(x)$ but it didnt get me anywhere.

I also tried defining $Ker_x = {vin mathbb{R}^n | D_f(x)v=0}$, and tried showing $Ker_xsubset Ker_a$:

Let $xin B(a,delta$), and $vin Ker_x$,

$$|D_f(a)v|=|D_f(a)v-D_f(x)v+D_f(x)v|leq|D_f(a)v-D_f(x)v| +|D_f(x)v|=|D_f(a)v-D_f(x)v|leqepsilon|v|$$

Where the last inequality is from the continuity of $D_f(x)$.The problem is however, $delta$ is not fixed if I want $|D_f(a)v|=0$ so I can`t see how to find a neighbourhood of $a$ out of these inequalities.

I`ll be glad if anyone can either help me fix my solution or suggest another way of approching this problem.

If $rk(D_f(a))=p$, this is equivalent to saying that there exists a $ptimes p$-minor $A_p(x)$ (submatrix) of $D_f(x)$ such that $g(a)=det(A_p(a))neq 0$, since $f$ is $C^1$, $D_f$ and henceforth $g$ are continuous,.Let $I$ be an open interval containing $det(A_p(a)$, but not $0$, $g^{-1}(I)$ is an open subset containing $a$.