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You should presuppose that the limit function is continuous at fisrt, otherwise, you can piece two "different" analytic functions together as the limit function.

Suppose with the presupposition in mind. In my opinion, if there are a sequence of uniformly converging piecewise analytic functions which do not converge to an analytic limit , then Runge's Theorem should play an important role in the construction of the counterexample.
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In my opinion, if there are a sequence of uniformly converging piecewise analytic functions which do not converge to an analytic limit , then Runge's Theorem should play an important role in the construction of the counterexample.