I am using an application made with ​​j-cook for a mobile app backend.
I need to see outside of joomla the THUMBNAIL file generated by upload an image, but it is corrupt.
As I can do?

It should work
Are you sure about your try ?

You must use a link using your FRONT index.php and not BACK :
administrator/index.php will not be accessible for public.

You can also use the direct link access, but not recommended for content files.
There are many ways.

Can you send me in private the external link you want to use ?

the frontend is not public, I just use it for my client to upload information to the database and to upload pictures.
When creating a thumbnail is perfect for me, because they have all the same size. But it is impossible to open.
I'm going crazy, at this moment the says me:
You do not have permission to access this document.
All directory in 755 and files in 644. I cant understand

Are you using direct file access ?
The thumbs are hidden files (starting with a dot)
Check permissions on those files.
Try to create another file in the same directory.
Check also the OWNER accesses. The thumbs are created by the www-data user (php user)

In case of indirect file access :
Even if your front is not public, are the guests allowed to call index.php?option=com_xxxx ?
If not, you can check ACL after the code returning file (in the main entry file, xxxx.php)

Hope it helps.

I have gotten display the image in the app, but I can not get to download the image to be stored on the phone, when the app is not connected

Maybe the headers are wrong, maybe some php errors.

Try to desactivate all php errors Because if php is outputing error or warning, then the headers will be already sent and so the image cannot display or download.
Second solution to test it : In the image class, exit(0) just BEFORE to send headers. I do not remember the function but you will find it is easy.
So you will see what is the php error.

Hope it helps.