- WELCOME TO A LESSON ON CONTINUITY.
THE GOALS OF THIS VIDEO WILL BE TO DEFINE CONTINUITY
USING LIMITS, DETERMINE IF A FUNCTION
IS CONTINUOUS OR DISCONTINUOUS AT A GIVEN POINT,
AND TO ALSO DETERMINE THE TYPES OF DISCONTINUITY.
WHEN YOU TOOK ALGEBRA, YOU MAY HAVE DETERMINED IF A FUNCTION
WAS CONTINUOUS BY DETERMINING IF YOU COULD SKETCH THE FUNCTION
WITHOUT LIFTING UP YOUR PENCIL.
IF YOU CAN SKETCH THE FUNCTION WITHOUT LIFTING UP YOUR PENCIL,
IT IS A CONTINUOUS FUNCTION.
IF YOU CANNOT SKETCH THE FUNCTION
WITHOUT LIFTING UP YOUR PENCIL, IT HAS SOME SORT
OF DISCONTINUITY AND IS NOT A CONTINUOUS FUNCTION.
WE WANT TO REVISIT THIS IDEA AND ALSO DETERMINE IF A FUNCTION
IS CONTINUOUS AT A SPECIFIC POINT.
LET'S TAKE A LOOK AT A DEFINITION OF CONTINUITY
INVOLVING LIMITS.
A FUNCTION F IS CONTINUOUS AT X EQUALS "A" IF IT SATISFIES
THESE THREE CONDITIONS.
F OF "A" EXISTS, THE LIMIT OF F OF X
AS X APPROACHES "A" EXISTS, AND ALSO THAT THE LIMIT
OF F OF X AS X APPROACHES "A" IS EQUAL TO F OF "A",
MEANING YOU CAN FIND THE VALUE OF THAT LIMIT BY PERFORMING
DIRECT SUBSTITUTION.
A FUNCTION IS CONTINUOUS OVER AN INTERVAL
IF IT IS CONTINUOUS AT EACH POINT IN THAT INTERVAL.
BEFORE WE LOOK AT SOME EXAMPLES, LET'S TALK ABOUT TWO TYPES
OF DISCONTINUITY.
HERE I HAVE TWO FUNCTIONS THAT ARE NOT CONTINUOUS.
THEY CONTAIN DISCONTINUITY.
NOW THIS FIRST FUNCTION, YOU CAN SEE,
IT'S REALLY ONLY DISCONTINUOUS AT THIS POINT.
IF I WAS TO TAKE THIS POINT AND MOVE IT BACK DOWN
INTO THIS POSITION, IT WOULD THEN
MAKE THE FUNCTION CONTINUOUS.
THIS IS CALLED REMOVABLE DISCONTINUITY.
F OF X CAN BE MADE CONTINUOUS BY APPROPRIATELY DEFINING
OR REDEFINING F OF C.
NOW, THIS FUNCTION COULD NOT BE MADE CONTINUOUS
BY REDEFINING A SINGLE POINT.
YOU CAN SEE HERE THAT IT HAS A VERTICAL ASYMPTOTE.
THIS IS CALLED NON-REMOVABLE DISCONTINUITY.
F OF X CANNOT BE MADE CONTINUOUS BY REDEFINING F OF C.
THIS TYPE OF DISCONTINUITY OCCURS WHEN THE LIMIT
DOES NOT EXIST AT C.
LET'S TAKE A LOOK AT AN EXAMPLE.
DETERMINE IF THE GIVEN FUNCTION IS CONTINUOUS
AT X = -2, 2, AND 4.
LET'S FIRST CONSIDER X = -2.
SO WE'RE LOOKING AT THE FUNCTION IN THIS REGION HERE.
AND, REMEMBER, WE WANT TO STATE IF IT IS DISCONTINUOUS,
WHICH IT LOOKS LIKE IT IS, WHAT IS THE REASON
BEHIND IT BEING DISCONTINUOUS.
SO I REWROTE THE THREE CONDITIONS DOWN HERE
THAT WE CAN VERIFY. FIRST, F OF "A" EXISTS.
WELL, IN THIS CASE, F OF "A" WOULD BE F OF -2
AND F OF -2 WOULD BE = -1 FROM OUR GRAPH.
SO IT SATISFIES THAT CONDITION OKAY.
THE NEXT CONDITION, THE LIMIT AS X APPROACHES "A" MUST EXIST.
SO WE'RE TRYING TO FIGURE OUT IF THE LIMIT
AS X APPROACHES -2 EXISTS.
REMEMBER, TO DETERMINE THIS LIMIT WE MUCH APPROACH -2
FROM BOTH SIDES.
THE RIGHT-SIDED LIMIT LOOKS LIKE IT'S APPROACHING 1.
THE LEFT-SIDED LIMIT LOOKS LIKE IT'S APPROACHING -1.
THEREFORE, THIS LIMIT DOES NOT EXIST.
THEREFORE, WE HAVE A FUNCTION THAT IS NOT CONTINUOUS
AT X = -2.
AND, IN FACT, THIS WOULD BE NON-REMOVABLE,
THIS CONTINUITY, BECAUSE WE COULD NOT MAKE IT CONTINUOUS
BY JUST REDEFINING THIS ONE POINT.
LET'S NOW CONSIDER THIS FUNCTION AT X = 2.
SO NOW WE'RE CONCERNED ABOUT THE FUNCTION AT THIS POINT.
SO, AGAIN, WE MUST SATISFY THESE THREE CONDITIONS.
AND RIGHT OFF, WE COULD SAY THAT F OF 2 DOES NOT EXIST.
THEREFORE, WE HAVE A FUNCTION THAT IS NOT CONTINUOUS AT X = 2,
BUT THE TYPE OF DISCONTINUITY HERE IS REMOVABLE.
IF WE WERE TO REDEFINE F OF 2, WE COULD MAKE
THIS FUNCTION CONTINUOUS.
TAKE A LOOK AT IT AT X = 4.
AT X = 4, LET'S SEE, DOES F OF 4 EXIST?
IT DOES.
IT LOOKS LIKE IT WOULD BE = 2.
NEXT, WE NEED TO CONSIDER THE LIMIT OF THIS FUNCTION
AS X APPROACHES 4 AND SEE IF THIS EXISTS.
AS WE APPROACH 4 FROM BOTH THE RIGHT SIDE
AND THE LEFT SIDE, WE CAN SEE THAT WE ARE GOING
TO BE APPROACHING THIS CLOSED POINT HERE.
THEREFORE THIS LIMIT WILL BE = 2.
SO WE VERIFIED F OF "A" EXISTS,
WE VERIFIED A LIMIT EXISTS, AND NOW WE NEED TO MAKE SURE
THAT WE CAN FIND THIS LIMIT BY DOING DIRECT SUBSTITUTION.
THE LAST THING WE NEED TO VERIFY IS THAT THE LIMIT CAN BE FOUND
BY PERFORMING DIRECT SUBSTITUTION,
IN THIS CASE, WHEN X IS 4.
SO NOTICE NOW THAT F OF 4 IS EQUAL TO 2 AND THE LIMIT
IS ALSO EQUAL TO 2.
THEREFORE THIS CONFIRMS THAT THIS FUNCTION
IS CONTINUOUS AT X = 4.
OKAY, I HOPE THAT HELPS EXPLAIN HOW TO DETERMINE
IF A FUNCTION IS CONTINUOUS AT A GIVEN POINT
AND ALSO TO DETERMINE WHAT TYPE OF DISCONTINUITY MIGHT EXIST.
THANK YOU FOR TUNING IN.