A little bit of research on Browser support for functions keys. The keys marked with OK are available to programmers, the rest is dedicated to the browser specific functionality.
OK - is available
N/A - not available
IE 9+:
F1 – **onkeydown: **OK, **onkeyup: **N/A (opens IE Help file in a new window)
F2 – **onkeydown: **OK, **onkeyup: **OK
F3 – **onkeydown: **OK, **onkeyup: **N/A (opens search box)
F4 – **onkeydown: **OK, **onkeyup: **N/A (opens url history)
F5 – **onkeydown: **N/A (refreshes the screen), **onkeyup: **OK
F6 – **onkeydown: **OK, **onkeyup: **N/A
F7 – **onkeydown: **OK, **onkeyup: **N/A (opens Caret browsing dialog)
F8 – **onkeydown: **OK, **onkeyup: **OK
F9 – **onkeydown: **OK, **onkeyup: **OK
F10 – **onkeydown: **OK, **onkeyup: **OK (+ enables/disables top task bar)
F11 – **onkeydown: **OK, **onkeyup: **OK (enables/disables full screen)
F12 – **onkeydown: **OK, **onkeyup: **N/A (enables/disables Inspect Element screen)
Chrome 33.0:
F1 – **onkeydown: **OK, **onkeyup: **N/A (opens Chrome Help file in a new tab)
F2 – **onkeydown: **OK, **onkeyup: **OK
F3 – **onkeydown: **OK, **onkeyup: **N/A (opens search box)
F4 – **onkeydown: **OK, **onkeyup: **OK
F5 – **onkeydown: **OK, **onkeyup: **OK (refreshes the screen)
F6 – **onkeydown: **OK, **onkeyup: **N/A
F7 – **onkeydown: **OK, **onkeyup: **OK
F8 – **onkeydown: **OK, **onkeyup: **OK
F9 – **onkeydown: **OK, **onkeyup: **OK
F10 – **onkeydown: **OK, **onkeyup: **OK (second press only – enables/disables full screen)
F11 – **onkeydown: **OK, **onkeyup: **OK (enables/disables full screen)
F12 – **onkeydown: **OK, **onkeyup: **N/A (second press only – enables/disables Inspect Element screen)
Firefox 19.0:
F1 – **onkeydown: **OK, **onkeyup: **OK
F2 – **onkeydown: **OK, **onkeyup: **OK
F3 – **onkeydown: **OK, **onkeyup: **OK (opens search box)
F4 – **onkeydown: **OK, **onkeyup: **OK
F5 – **onkeydown: **N/A, **onkeyup: **OK (refreshes the screen)
F6 – **onkeydown: **OK, **onkeyup: **N/A
F7 – **onkeydown: **OK, **onkeyup: **OK (+ opens Caret browsing dialog)
F8 – **onkeydown: **OK, **onkeyup: **OK
F9 – **onkeydown: **OK, **onkeyup: **OK
F10 – **onkeydown: **OK, **onkeyup: **N/A (second press only – enables/disables top task bar)
F11 – **onkeydown: **OK, **onkeyup: **OK
F12 – **onkeydown: **OK, **onkeyup: **OK