Mathematics

Problem

Consider the function

g(x) = \begin{cases} \dfrac{\sin(\pi x)}{x} + \dfrac{\lfloor x \rfloor}{1+x^{2}}, & x \neq 0, \\ b + \dfrac{\lfloor 0 \rfloor}{1+0^{2}}, & x = 0, \end{cases}

where

b

is a constant and

\lfloor x \rfloor

denotes the greatest integer less than or equal to

x

. (a) Identify all values of

x

at which

g

is discontinuous. For each such

x

, classify the discontinuity as removable, jump, or infinite and provide supporting justification. (b) Determine the value of

b

that removes any removable discontinuity at

x = 0

. (c) Let

\tilde{g}

denote the function obtained by substituting the value of

b

found in part (b). Prove, using the

\varepsilon

–

\delta

definition of limit, that

\tilde{g}

is continuous at

x = 0