Consider the function g(x)=xsin(1/x) on the open interval (0,1). The function is not defined at x=0, and we do not extend its definition to that point.
(a) Determine whether g is bounded on (0,1). Support your answer with reasoning.
(b) Decide whether the Extreme Value Theorem guarantees the existence of an absolute maximum or minimum of g on (0,1). Provide a complete justification.
(c) Evaluate x→0+limg(x) and explain how this limit informs your conclusion in part (b).
(d) State whether g actually attains an absolute maximum or absolute minimum value on (0,1). Justify your answer thoroughly.