Analytic continuation from a continuum to its neighborhood

Let $r$ be a positive number. A function $f$ analytic in an open set $\mathcal O\subset\mathbb C$ is called $r$-analytic on the set $E$, $E\subset\mathcal O$, if $\varlimsup_{k\to+\infty}\bigl|\frac{f^{(k)}(t)}{k!}\bigr|^{1/k}\le\frac1r$ ($t\in E$).
THEOREM. Let $K$ be a compact connected subset of the plane. For any $r>0$ there exists an open neighborhood $V$ of the set $K$ such that any function $r$-analytic on coincides in some neighborhood of the set $K$ with a function analytic in $V$.
This theorem answers a question posed in the collection (RZhMat., 1979, 3B536, pp. 33–35 of the book).