Is the rank of a differential of a transformation continuous?

Question

let $T$ be a continuously differentiable transformation from $3$-space to $3$-space.

then clearly, $J(p)$ which is the jacobian at point $p$ is continuous because each component in $dT$is continuous as given in the question. But what about the rank of $dT$? Seems like it is not , but can someone give a proof?

Answer 1

Consider $T(x,y,z)=(x^2,y^2,z^2)$. The rank of its differential is discontinuous at $0$.