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#Wikis_and_Encyclopedias for other definitions of "piecewise continuously differentiable" and found none.

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

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

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

- Complex Made Simple, 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 $x_i$.

- Mathematics in Population Biology, 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}.

- Analysis II, Herbert Amann,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 ($x_{i−1},..x_i$).

- A First Course in Harmonic Analysis, Anton Deitmar:

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

3. 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).

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

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


 * (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∈{1,…,n}.