About smoothness of pasted function - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T18:28:30Z http://mathoverflow.net/feeds/question/109478 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/109478/about-smoothness-of-pasted-function About smoothness of pasted function user 123 2012-10-12T18:16:44Z 2012-10-13T10:07:45Z <p>Let $M\subset \mathbb R^2$ be a halfplane and $X$ be a Banach space. </p> <p>We say that a function $f:M\rightarrow X$ is a of class $C^\infty$ on $M$ if it is of class $C^\infty$ on $int M$, for each multiindex $\alpha$ and for each $x$ from boundary of $M$ there exists $\lim_{y\rightarrow x \atop y\in int M} D^\alpha f(y)$, and function $D^\alpha f$ defined on $int M$ in usual sense and for $x$ from boundary by $D^\alpha f(x) =\lim_{y\rightarrow x \atop y\in int M} D^\alpha f(y)$ is continuous.</p> <p>Let $M_1=(-\infty, 0] \times \mathbb R$, $M_2=[0,\infty) \times \mathbb R$.</p> <p>Let's assume that functions $f:M_1\rightarrow X$, $g: M_2 \rightarrow X$ are of class $C^\infty$. </p> <p>Is it the function $h:\mathbb R^2 \rightarrow X$ defined by $h(x,y)=f(x,y)$ for $(x,y) \in M_1$ and $h(x,y)=g(x,y)$ for $(x,y)\in M_2$ of clas C^\infty$?</p> <p>Edit:</p> <p>I forgot to add that for each$(x,y)$from the boundary of M we assume$f(x,y)=g(x,y)$and for each boundary point$(x,y)$and each multindex$\alpha$we assume that$D^\alpha f(x,y)=D^\alpha g(x,y)\$.</p>