[Phys-l] continuous but not differentiable

[John Denker <jsd@av8n.com>]

On 03/30/2012 08:59 AM, Bernard Cleyet wrote:
> every thing is linear in a narrow range unless poorly behaved (discontinuous).

I hope this doesn't seem unduly nitpicky ... but "discontinuous" 
functions are not the only ones that fail to have a linear 
approximation.

As a familiar example, the absolute value function is 
continuous at the origin, but has no linear approximation 
there, as is obvious from the fact that it is not 
differentiable.

This point is not some irrelevant mathematical nitpick.
Functions that have no Taylor-series expansion show up in 
physics, for example in connection with critical phenomena.
  http://www.nobelprize.org/nobel_prizes/physics/laureates/1982/

Continuing down that road, the Weierstrass function is 
everywhere continuous but nowhere differentiable!