Modulus of continuity of piecewise analytic functions (Q1810275)

Conditions under which the modulus of continuity \(\omega(f; \delta)\) of a piece-wise real-analytic function \(f : [a, b] \rightarrow {\mathbb R}\) becomes analytic at zero are found. The results obtained are of the following type. Theorem 1. Let \(f\) be piece-wise real-analytic on \([a, b]\). If \[ \sup_{x\in D_{N}} d(x) < \sup_{x\in M\setminus D_{N}} d(x), \] then \(\omega(f; \delta)\) is an analytic function at zero. In the case of the opposite inequality \[ \sup_{x\in D_{N}} d(x) > \sup_{x\in M\setminus D_{N}} d(x) \] \(\omega(f; \delta)\) is not an analytic function at zero. Here \[ d(x) = \begin{cases} d_{lx},& x\in M_l \setminus M_r, \\ \max \{d_{lx}, d_{rx}\},& x\in M_l \cap M_r, \\ d_{rx},& x\in M_r \setminus M_l, \end{cases} \] \(d_{lx_{0}}\), \(d_{rx_{0}}\) are multiplicities of zeros at a point \(x_{0}\) of the functions \(f^{\prime}_{l}(x) - f^{\prime}_{l}(x_{0})\), \(f^{\prime}_{r}(x) - f^{\prime}_{r}(x_{0})\), respectively, and \(f^{\prime}_{l}\), \(f^{\prime}_{r}\) are left-, right-sided derivatives, \[ M_l = \left\{x\in [a,b] : \left| f^{\prime}_{l}(x)\right| =m\right\},\, M_r = \left\{x\in [a,b] : \left| f^{\prime}_{r}(x)\right| =m\right\},\, M = M_{l}\cup M_{r}, \] \[ m = \sup_{x\in [a,b]} \begin{cases} f^{\prime}_{l}(x), & x = a, \\ \max\{f^{\prime}_{l}(x), f^{\prime}_{r}(x)\}, & a < x < b, \\ f^{\prime}_{r}(x), & x = b, \end{cases} \] the set \(D_{N}\) consists of points \(x\in M_{l}\cap M_{r}\) for which non of the ``numbers'' \(d_{lx}\), \(d_{rx}\) divides the other ``number'' and \(f^{\prime}_{l}(x) = f^{\prime}_{r}(x)\). Characterization of the non-analytic modulus of continuity for piecewise real-analytic functions is given, too.