Is the map $\Phi: \mathbb{R}^{N\cdot m}\times M\ni (\lambda_{p,i},x) \mapsto f(x)+\sum_{p}\sum_{i=1,...,m}\lambda_{p,i}x_{p,i}(x)\cdot \phi_{p}(x) \in \mathbb{R}$ globally defined, where M is compact smooth manifold, $x_{p,1},...,x_{p,m}\ $ are local coordinates,$\phi_{p}$ are partition of unity for chart at p and f $\in C^{\infty}(M,\mathbb{R})$?
Are there any well-defined problems in the intersection of coordinate charts covering M? Since M is a manifold ,the coordinate charts will agree on the intersection.
This is for proving Morse functions are generic.
The problem was that I had to multiply by partition of unity functions. That fixes the problem of making it globally defined.