Let's say that $f(x, y) $ is a real valued continuous function which happens to have continuous first derivatives with respect to $x $ and $y $ and that it admits continuous second partial derivative with respect to $y $ except at most countably many points at which nevertheless it admits finite left- and right-second derivative.
Is it possible to approximate it using functions like $$ f_n(x, y) = \sum_{k=0}^n g_{n,k}(x)h_{n,k}(y) $$
where each of $g_{n,k}(\cdot) $ is continuously differentiable and $h_{n,k} $ is continuously differentiable and twice continuously differentiable except on a countable set of points in the domain of definition where it has finite right and left-second derivatives ?
I do believe the answer to be yes, but I wonder if the conditions are straightforward. I am looking at this with the intent to apply it in the proof of a more general Ito-formula for Brownian motion and time without assuming that the original $f $ is twice continuously differentiable with respect to $x. $
Thank you.
Maurice