Differentiable function - definition on a manifold

[cianfa72]

Hi,

a basic question related to differential manifold definition.

Leveraging on the atlas's charts $\left\{(U_i,\varphi_i)\right\}$ we actually define on $M$ the notion of differentiable function. Now take a specific chart $\left(U,\varphi \right)$ and consider a function $f$ defined on it that happens to be not differentiable in that specific chart.

My question is: for the given function defined on $M$ could it be the case in resulting differentiable when represented in another atlas's chart (e.g. in the $\left(V,\gamma \right)$ chart supposed to be compatible with $\left(U,\varphi \right)$ )?

In other words: we know the notion of differentiable function is compatible across charts belonging to the same differential structure, but what about a not differentiable function as represented in one of the charts ?

Thanks

 

[martinbn]

In other words: we know the notion of differentiable function is compatible across charts belonging to the same differential structure, but what about a not differentiable function as represented in one of the charts ?

If it were differentiable in one chart, then it would be in all charts (according to the first part of your sentence). So, if it isn't in one chart, it will not be in the others.

 

[cianfa72]

Sure, thank you!