I would like to update (change the content) some part of the webpage without updating the whole page (without refreshing the page). Of course first thing you would write as an answer will be "Ajax!!!", but I know that search engines will not be able to get the content of that part if I do with Ajax.
So is there any method or workaround to update page partly still remaining that part visible to the search engines?
In short: No.
The moment you start breaking the "One URL" = "One Page" principle, you alienate search engines (and bookmarkers, and people who want to send a link to a friend).
The other main option is frames (which replaces "invisible to search engines" with "orphan pages").
A third option is stuffing all the content on to the page in the first place, and then using JS to set class names on elements to hide some content.
In general, sharing duplicate meta-content (navigation etc) between pages is the cleanest, simplest, most reliable approach.
yes.. it is possible by not using ajax ;)
what you can do is load all content that is necessary for displaying. When the page is fully loaded you hide the content that should not be visible at startup. This can be done by CSS or javascript. When the content should be loaded by pressing a button you can alter the css property (with javascript) of the element so it is visible
I would put in normal links to normal pages and have javascript replace the link with the ajax link.
That way you are also providing for the people who have javascript disabled.