Use SelectedIndexChanged event to do this.
Set autopostback true for continent and country dropdown
Suppose there are three method
ShowContinent(),
ShowCountry() and
ShowCity()
Use ShowContinent() on pageload, ShowCountry() on Continent dropdown's SelectedIndexChanged event and ShowCity() on Country dropdown's SelectedIndexChanged event
For stop page refresh use ajax, put all three dropdown inside
ContentTemplate tag
<updatepanel id="UpdatePanel1" runat="server">
<contenttemplate>
</contenttemplate>
</updatepanel>