First let me state a result of Kazdan and Warner

Let $M$ be a compact orientable two dimensional manifold. Let $f:M \rightarrow \mathbb{R}$ be a function that has the same sign as the Euler characteristic of $M$ at some point $p$. Then $M$ admits a Riemannian metric such that the Gaussian Curvature is equal to $f$.

I want to know if there is a combinatorial analogue of this statement. Namely, is the following statement true:

Let $M$ be a compact orientable two dimensional manifold. Let $f:M \rightarrow \mathbb{R}$ be a function that is zero except at a finite set of points $p_1 \ldots p_n$. At one of those points, $f$ has the same sign as the Euler characteristic of $M$. Then does $M$ admit a triangulation with $p_1 \ldots p_n$ as vertices such that the ``Curvature'' at each of these points is same as $f$?

Here by curvature we mean the angle deficit from $2 \pi$. And the curvature of $M$ at any point that is not a vertex is defined to be zero.

I understand from the counter example Liviu gave that I have to relax my definition of triangulation. Let me explain the motivation for my question. What I was wondering is that can one hope to prove the Kazdan Warner theorem this way.... .....the basic idea is this.......given a smooth function $f$ that satisfies the Gauss Bonnet sign condition, construct a sequence of discrete functions $f_n$ (which are zero at all but finitely many points) that converge in some appropriate sense to $f$ (pointwise?) Then argue that the singular metrics $g_n$ converge in some sense to the desired solution. Is there any chance such an approach can be made precise? To start with one can just ask for solutions in a ``weak'' sense.

Another question I have is......are there any questions in geometric analysis...particularly questions of this nature ``find a metric satisfying some condition'' where such an approach actually works.......ie first solve a
discretized version of the problem and then show that the desired solution is in some sense an appropriate limit?

A particular example I had in mind while asking this question is the following : A proof that the De Rham Cohomolgy is isomorphic to the singular cohomology using combinatorial hodge theory. The idea of the proof roughly is that one defines a discrete analogue of the $d$ operator and discretizes the De Rham Cohomology and shows that in an appropriate sense this converges to the singular cohomology (and hence giving the isomorphism).

Of course this particular example has nothing to do with finding metrics on manifolds, but I stated that example to illustrate that general idea......you prove some statement by solving a discrete version and then try to show the desired solution is some appropriate limit. Are there other instances where such an idea has been carried out (particularly for the type of questions I am asking .......finding metrics satisfying some condition ....or may be even isometric embedding problems)?