Mathematics

Problem

Construct a function

k

defined on the interval

-3\le x\le 3

whose graph satisfies all the following conditions: 1.

k(0)=2

. 2.

\displaystyle \lim_{x\to 0^-}k(x)=1

. 3.

\displaystyle \lim_{x\to 0^+}k(x)=3

. 4.

\displaystyle \lim_{x\to 2}k(x)=4

but

k(2)

is not defined. 5.

k

is continuous at every real number other than

x=0

and

x=2

. a. (i) On graph paper, draw one possible graph of

k

that meets all the above conditions for

-3\le x\le 3

. Clearly indicate open and closed points. b. (ii) For each of the points

x=0

and

x=2

, state the left–hand limit, right–hand limit (if applicable), the value of the function (if it exists), and explain why

k

is or is not continuous there. c. (iii) Briefly describe how you ensured continuity of

k

at all other points.