Mathematics

Problem

(a) Define continuity of a real-valued function at a point

x=c

. (b) A function

f

is given by

f(x)=\begin{cases} g(x), & x<c,\\[4pt] h(x), & x\ge c, \end{cases}

where

g

and

h

are real-valued functions. Write the explicit algebraic conditions that must hold at

x=c

for

f

to be continuous there. (c) Suppose

g

and

h

are polynomials and the symbol

k

appears as an unknown constant only in

g

. Explain, without performing any numerical calculation, why solving the continuity conditions at

x=c

can determine at most one real value of

k

f(x)=\begin{cases} g(x), & x<c,\\ h(x), & x\ge c, \end{cases}