# Maximum modulus principle for continuous functions

What is the reason maximum modulus principle will fail for non-holomorphic functions in the closed disk? I am reading the part in Rudin on Poisson summation formula and I was wondering what a good counterexample would be like. I am assuming $f$ to be continuous and is bounded on the unit disk, so functions like $\log[x],\frac{1}{x}$ are excluded.

Rudin proved this holds for all polynomial functions, and it is not too difficult to extend it to all holomorphic functions on the unit disk via classical arguments. So i am thinking about given a function on the boundary of the circle and run a heat equation on it, or given a heat distribution in the middle circle of the disk instead. So in the extreme case I would have a function has value 1 on $0$, and 0 on the circle. This contradicts the maximum modulus principle - but why can't we approximate this simple function via holomorphic functions and use dominated convergence theorem? In other words, what is the reason that $C_{0}(D^{2})$ is not a completion of $H(D^{2})$?

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For the second question, it's not difficult to show that if $f_j$ are holomorphic and converge locally uniformly to a function $f$, then $f$ is also holomorphic. (This follows, for example from Morera's theorem.) Even if you just assume that the $f_j$:s converge pointwise, there is a theorem by Osgood showing that limit function is holomorphic on an open, dense set, so we are very far from being able to approximate arbitrary continuous functions by holomorphic ones.