Gerald Edgar: thank you, but I am now pretty convinced it is the other one (the one I give). Unfortunately I did not remember dr's MO name right away, but then remembered a somewhat recent meta thread http://tea.mathoverflow.net/discussion/1284/is-this-question-appropriate-on-mo/

(What question are you talking about?)

Edit: seems to be that one http://mathoverflow.net/questions/87415/how-to-construct-a-function-belonging-to-a-class-of-functions-f-colon-0-1-tim

Brief response: the question is not selfcontained, which is a minor issue but might explain why it gets little attention. Also it is not exactly clear what type of singularities you want. You seem to want that the k+1 th derivative 'blows up'. But what does this mean precisely if the original function had two variables.

Moreover, and very importantly, it is not clear why you want this. In the 1 dim case you had a different type of singularity. Now you generalize two two dim and want a different singularity. So this seems not very well motivated (without further elaboration).