Mathematics

Problem

A function

f

is described as follows. • For

x<1

, the graph is the horizontal line

y=2

. • At

x=1

, there is an open circle at

(1,2)

and a filled circle at

(1,3)

. • For

x>1

, the graph is the line

y=x+1

. Answer the following questions about

f

. a. (i) Find the left–hand limit

\displaystyle \lim_{x\to 1^-}f(x)

. b. (ii) Find the right–hand limit

\displaystyle \lim_{x\to 1^+}f(x)

. c. (iii) Does

\displaystyle \lim_{x\to 1}f(x)

exist? Give a reason. d. (iv) Is

f

continuous at

x=1

? Justify your answer with reference to the definition of continuity.