Definition talk:Piecewise Continuously Differentiable Function

Other definitions of piecewise continuously differentiable
1. I have searched the list at http://www.proofwiki.org/wiki/ProofWiki:Community_Portal#Magazines for other definitions of piecewise continuously differentiable and found:

- Agarwal and O’Regan :
 * (1) replaced by: $f$ is piecewise continuous (according to Definition 11.1 in the book).
 * (2) replaced by: $f'$ is piecewise continuous (according to Definition 11.1 in the book), which means that $f'$ is not required to be continuous or defined at $x_i$ for every $i$∈{0,…,n}.
 * Term used: piecewise smooth. (I used the search function of maa.org and used the search term "piecewise continuous".)
 * Ivar Sand (talk) 09:37, 12 August 2013 (UTC)

- Kaplan :
 * (1) replaced by: $f$ is piecewise continuous.
 * Term used: piecewise smooth. (I used the search function of maa.org.)
 * Ivar Sand (talk) 09:37, 12 August 2013 (UTC)

2. I have searched the list at http://www.proofwiki.org/wiki/ProofWiki:Community_Portal#Wikis_and_Encyclopedias for other definitions of "piecewise continuously differentiable" and found none.

3. I have found these on the Internet (I have done only a limited search):

- In Methods of Mathematical Physics, Differential Equations by Richard Courant and D. Hilbert :


 * (2) is replaced by: The derivative of $f$ is a piecewise continuous function. Ivar Sand (talk) 10:27, 24 July 2013 (UTC)

- In Complex Made Simple by David C. Ullrich :


 * [$x_{i−1}..x_i$] in (2) replaced by ($x_{i−1}..x_i$).
 * $f'$ has one-sided limit(s) at every $x_i$.

- In Mathematics in Population Biology by Horst R. Thieme :


 * [$x_{i−1}..x_i$] in (2) replaced by ($x_{i−1}..x_i$).
 * Observation: $f'$ is allowed to exist but be discontinuous at some point $x_i$ where i∈{1,…,n-1}.

- In Analysis II by Herbert Amann and Joachim Escher :


 * (1) is replaced by: $f$ is piecewise continuous,
 * $f$ is continuously differentiable on [$x_{i−1}..x_i$] in (2) replaced by $f'$ is uniformly continuous on ($x_{i−1}..x_i$).

- In A First Course in Harmonic Analysis by Anton Deitmar :


 * (This seems not to be a different definition, only a reformulation). Ivar Sand (talk) 08:24, 26 July 2013 (UTC)

4. I have searched the list at http://www.proofwiki.org/wiki/ProofWiki:Community_Portal#Wikis_and_Encyclopedias for other definitions of "piecewise continuously differentiable" by searching for "piecewise smooth", which is sometimes synonymous with "piecewise continuously differentiable" and found:

- scholarpedia.org :


 * [$x_{i−1}..x_i$] in (2) replaced by ($x_{i−1}..x_i$).

- planetmath.org :


 * (This seems not to be a different definition, only a reformulation).

5. I have found these on the Internet (I have done only a limited search):

In Linear Partial Differential Equations for Scientists and Engineers (2007) by Tyn Myint-U and Lokenath Debnath: :


 * (1) is replaced by: $f$ is piecewise continuous,


 * [$x_{i−1}..x_i$] in (2) is replaced by ($x_{i−1}..x_i$),


 * included in (2): the one-sided limits $f'(x_{i−1}+)$ and $f'(x_i-)$ exist for every $i \in \{1, \ldots, n\}$.