Subject: Is taking the complete dose of an antibiotic really needed?

If I understand correctly that the bacteria can only change by
 mutation and the antibotic kills some of the bacteria but not all and 
continued growth of the bacteria gives an increased chance of mutation.
Then,if the mutation is resistant to the antibiotic a new and resistant
bacterial strain is born. This seems to have nothing to do with 
whether a person completes the prescribed dose of antibiotic. 
The only way to stop the uncontrolled mutations in bacteria 
appears to be in discontinuing the use of antibiotics in medicine.
Not an attractive alternative. Is the above correct or did I miss something in Bio.? 

Re: Is taking the complete dose of an antibiotic really needed?