Front Derailleur stays on smallest chain ring
If this is an older bike with a standard friction shifter, you need to adjust the friction.



On the shaft where the lever pivots is an adjustment knob of some sort.  Simply tighten that knob until the lever is "stiff" operating but can still be moved with reasonable ease.

If, on the other hand, you have an indexed shifter then the procedure to correct this problem depends on the specific type you have (and how repairable that particular type is).  A picture of your shifter would be most helpful if it is not the standard friction style.