Quasiconcave, Continuous, Monotone Functions

Question

In another thread a fellow user found a counter example to a monotonic multivariate function that is not quasi concave.

My question: if we add a continuity assumption - is any continuous, monotonic (gradient greater than 0) multivariate function quasi concave?

Thank you!

Answer 1

If I took the right definitions then

$$f(x,y) = \max\{x,y\}$$

might be a counterexample.