The 80/40 switch on the Apple IIc was designed for customers who use their television set as a monitor. A television set doesn't display 80 columns of text very legibly; 40 columns reads much better. Unfortunately, most Apple II software packages, written before the Apple IIc, don't check the bit set by the 80/40 switch.
Because the bit setting is independent from the PR#3 command, programmers can use the bit set by the 80/40 switch for features other than 80/40 column display:
10 TEXT:HOME:REM INITIALIZATION
100 SWITCH = PEEK(49248):REM MEMORY LOCATION FOR CURRENT SWITCH SETTING
110 VTAB 12
120 IF SWITCH > 127 THEN PRINT "80/40 SWITCH DOWN":REM OR ANYTHING ELSE
130 IF SWITCH < 128 THEN PRINT "80/40 SWITCH UP ":REM OR ANYTHING ELSE
140 GOTO 100
For more info, see the two-volume Apple IIc (16K ROM) Technical Reference Manual, pages 5 and 77 of Volume I and page 128 of Volume II, or the one-volume manual published by Addison-Wesley, pages 5, 80, and 342.