Consider the unit ball $\Delta=\{|z|<1\}$ and a Lipschitz (meaning that it is the graph of some Lip. real function) segment, say $\Gamma=[1,7]$.
Consider $f$ holomorphic in a neighborhood of the ball, for convenience we take $f\in\mathcal O(\Delta_4)$ and continuous on the whole plane $\Bbb C$.
I want to approximate $f$ on $\Delta\cup\Gamma$ by entire functions.
Call $g$ the Taylor polynomial of $f$ centered in $0$ and truncated at a sufficiently high order: it is entire and approximate $f$ on $\Delta_4$ (the open ball of radius 4).
Call $h$ the entire function (provided by a theorem by P.Manne contained in his phd thesis, I cannot find online; the important part is that such a function exists) approximating $f$ on $\Gamma$.
The problem is that we don't know how does $g$ behaves on $\Gamma$ and viceversa, we don't know how does $h$ behaves on $\Delta$.
Defining a smooth function $\chi(z)$ to be 0 on $|z|<2,|z|>8$ and $1$ on $3<|z|<7$ and considering $$ \widetilde f = g+\chi\cdot(h-g) $$ this is continuous (and with compact support) on the whole plane and approximate $f$ on $\Delta\cup\Gamma$.
If allowed to use Mergelyan theorem I could approximate $\chi$ with an entire function on $|z|<9$ and conclude, but I can't.
Any hint on how to glue two entire functions without using Mergelyan thm?
