Auto detect and select local city and keep the setting while surfing the site

I’ve chose a default city for my installation but instead I would like my site to auto detect city based on location of the visitor.

Second question is how can I make sure the selected city stays selected while surfing the site? Right now I can select a city in the drop down but when I surf around on the city the dropdown just goes back to “change location” instead of showing the city I chosed.