Smoothness repaired by iteration (Q2350282)

Piecewise \({{C}^{1}}\) self-mappings \(f\) of \([0,1]\) having only one non-smooth point \({{x}_{0}}\in (0,1)\) are considered. Let \({{f}_{1}}=f{{|}_{[0,{{x}_{0}})}}\) and \({{f}_{2}}=f{{|}_{({{x}_{0}},1]}}\). Three cases will arise: 1) \({f}'\) has a jump discontinuity at \({{x}_{0}}\); 2) \({{f}_{1}}^{\prime }\) and \({{f}_{2}}^{\prime }\) are both bounded but at least one of \({{\lim }_{x\to {{x}_{o}}-}}{{f}_{1}}^{\prime }\left( x \right)\) or \({{\lim }_{x\to {{x}_{o}}+}}{{f}_{2}}^{\prime }\left( x \right)\) fails to exist; 3) at least one of \({{f}_{1}}^{\prime }\) or \({{f}_{2}}^{\prime }\) is unbounded. Necessary and sufficient conditions for \(f\circ f\) to be \({{C}^{1}}[0,1]\) smooth are given for the first two cases. In the third case, necessary conditions for \(f\circ f\) to be \({{C}^{1}}[0,1]\) smooth and sufficient conditions under which \(f\circ f\) is not \({{C}^{1}}\) are given.