- NOW WE'LL LOOK AT THREE EXAMPLES
OF CONTINUITY AT A POINT.
BASED UPON THE GRAPH, WE WANT TO SELECT
ALL OF THE STATEMENTS THAT ARE TRUE.
WE FIRST WANT TO DETERMINE IF F OF -3 IS DEFINED.
SO LOOKING AT OUR GRAPH, NOTICE HOW HERE IS WHERE X = -3.
NOTICE HOW AT -3 WE HAVE AN OPEN POINT HERE
AND WE HAVE A CLOSED POINT HERE, WHICH MEANS THE FUNCTION
IS DEFINED AT X = -3.
AND THE FUNCTION VALUE WOULD BE -5.
SO F OF 3 IS DEFINED IS TRUE. WE'LL GO AHEAD AND CHECK THAT.
NEXT, THE LIMIT AS X APPROACHES -3
OF F OF X EXISTS.
IN ORDER FOR THIS LIMIT TO EXIST,
AS WE APPROACH -3 FROM THE LEFT AND THE RIGHT,
WE MUST BE APPROACHING THE SAME FUNCTION VALUE.
SO NOTICE AS WE APPROACH -3 FROM THE RIGHT
OR FROM THE POSITIVE SIDE, IN THIS DIRECTION HERE,
WE'RE APPROACHING A FUNCTION VALUE OF 5.
BUT AS WE APPROACH -3 FROM THE NEGATIVE SIDE,
OR LEFT SIDE IN THIS DIRECTION, WE'RE APPROACHING
A FUNCTION VALUE OF 2.
SO BECAUSE WE'RE NOT APPROACHING THE SAME FUNCTION VALUE
FROM BOTH SIDES, THIS LIMIT DOES NOT EXIST.
SO, THIS STATEMENT IS FALSE.
AND THE NEXT STATEMENT SAYS THE LIMIT AS X APPROACHES
-3 OF F OF X = F OF -3.
WELL, IT CAN'T EQUAL F OF -3 IF THE LIMIT DOESN'T EXIST.
SO THAT'S NOT TRUE.
AND THESE LAST TWO STATEMENTS GO TOGETHER, EITHER F OF X
IS CONTINUOUS AT X = -3 OR IT'S NOT CONTINUOUS AT X = -3.
AND BY DEFINITION, ALL THREE OF THESE FIRST STATEMENTS
MUST BE TRUE IN ORDER FOR A FUNCTION
TO BE CONTINUOUS AT A POINT, OR IN THIS CASE, X = -3.
AND SINCE THESE TWO STATEMENTS ARE NOT TRUE, THE FUNCTION
IS NOT CONTINUOUS AT X = -3.
SO THESE ARE THE ONLY TWO TRUE STATEMENTS
BASED UPON THIS GRAPH.
LET'S TAKE A LOOK AT ANOTHER EXAMPLE.
DIFFERENT FUNCTION, SAME QUESTIONS.
AND NOW WE'RE CONSIDERING THE FUNCTION AT X = 4.
SO, FIRST, OF 4 IS DEFINED.
LOOKING AT THE GRAPH, HERE'S WHERE X = 4.
NOTICE HOW WE HAVE AN OPEN POINT HERE
AND A CLOSED POINT HERE, WHICH MEANS F OF 4 IS DEFINED
AND IT'S ALSO EQUAL TO 2.
SO THIS FIRST STATEMENT IS TRUE.
NEXT, THE LIMIT AS X APPROACHES 4 OF F OF X EXISTS,
SO WE'LL APPROACH 4 FROM THE RIGHT AND FROM THE LEFT.
WE'RE APPROACHING FROM THE RIGHT FROM THIS DIRECTION.
WE'RE APPROACHING FROM THE LEFT FROM THIS DIRECTION.
NOTICE FROM BOTH SIDES WE'RE APPROACHING
THIS FUNCTION VALUE HERE OR THE Y VALUE OF -4.
SO THE LIMIT DOES EXIST.
NEXT, WE HAVE THE LIMIT AS X APPROACHES 4
OF F OF X = F OF 4, MEANING WE CAN DETERMINE
THIS LIMIT BY PERFORMING DIRECT SUBSTITUTION
INTO THE FUNCTION. AND THIS IS NOT TRUE.
WE SAW EARLIER THAT F OF 4 IS EQUAL TO 2,
BUT THE LIMIT AS X APPROACHES 4 OF F OF X IS EQUAL TO -4.
SO WE CAN NOW DETERMINE THIS LIMIT BY PERFORMING
DIRECT SUBSTITUTION. THIS STATEMENT IS FALSE.
AND THEREFORE SINCE ALL THREE OF THESE ARE NOT TRUE,
THAT MEANS THE FUNCTION F OF X IS NOT CONTINUOUS AT X = 4.
SO THIS STATEMENT WOULD ALSO BE TRUE.
LET'S TAKE A LOOK AT ONE MORE EXAMPLE.
WE HAVE THE SAME STATEMENTS.
FIRST WE WANT TO DETERMINE IF F OF 2 IS DEFINED.
LOOKING AT OUR GRAPH, NOTICE HERE'S WHERE X = 2.
THE FUNCTION IS DEFINED AND F OF 2 DOES EQUAL -3.
SO, THIS FIRST STATEMENT IS TRUE.
NEXT, THE LIMIT AS X APPROACHES 2 OF F OF X EXISTS.
NOTICE AS WE APPROACH X = 2 FROM THE RIGHT
AND FROM THE LEFT, WE ARE APPROACHING
THE SAME FUNCTION VALUE, WHICH IS -3,
WHICH MEANS A LIMIT DOES EXIST.
NEXT, THE LIMIT AS X APPROACHES 2 OF F OF X = F OF 2,
MEANING WE CAN DETERMINE THIS LIMIT BY PERFORMING
DIRECT SUBSTITUTION, WHICH IN THIS CASE IS TRUE.
NOTICE HOW F OF 2 IS EQUAL TO -3,
WHICH IS ALSO THE SAME AS A LIMIT AS X APPROACHES 2.
AND SINCE ALL THREE OF THESE STATEMENTS ARE TRUE,
WE CAN SAY THAT F OF X IS CONTINUOUS AT X = 2.
SO I'LL LEAVE YOU WITH THE THREE CONDITIONS
THAT MUST BE MET IN ORDER FOR A FUNCTION
TO BE CONTINUOUS AT A POINT. I HOPE THIS WAS HELPFUL.