Let us say $x$ varies over the reals and I have a matrix $T$ which is a function of $x$. (The entries of $T$ depend on $x$)
Then I can enter such a matrix to some software and ask it to calculate the rank and the size of its kernel. The software promptly gives me back some integers as answers.
It seems that for $x=0$ I get a certain value of the rank and kernel dimension and for all other non-zero values of $x$ I get a different fixed set of numbers.
Is the above a generic case. Won't in general the kernel size and the rank depend non-trivially on the parameter?
If this behaviour (i.e a certain answer for $x=0$ and another answer for all other values of $x \neq 0$) is the generic behaviour then how does one prove such a thing?
It seems that the computer software (Mathematica) that I have been using gives the answer for $x \neq 0$ when I ask it to do the calculation for general $x$ and I need to check the $x=0$ case separately.
Is something like this known or can be known that the rank and kernel dimension of a parametrized matrix will vary smoothly with the parameter? (..obviously since the rank and kernel dimension are integers the only way they can depend nicely on the parameter is by being constant!..)
I would like to know if there are other caveats in general about doing such a calculation.