(a) Sketch the graph of a function on [-1,2] that has an...

(a) Sketch the graph of a function on $[-1,2]$ that has an absolute maximum but no absolute minimum.
(b) Sketch the graph of a function on $[-1,2]$ that is discontin- uous but has both an absolute maximum and an absolute minimum.

Key Concepts

Absolute Extremum

Absolute extrema refer to the highest and lowest function values attained on a given domain. An absolute maximum is the largest output value, while an absolute minimum is the smallest. Understanding these concepts is crucial for analyzing the overall behavior of functions and determining the range of outputs, especially when graphs or sketches are involved.

Extreme Value Theorem

The Extreme Value Theorem states that a continuous function on a closed interval will always have both an absolute maximum and an absolute minimum. This theorem is key to recognizing that if a function on a closed interval lacks one of these extrema, then it must be discontinuous. Thus, it guides the construction or analysis of examples that intentionally violate its conditions.

Continuity and Discontinuity

Continuity ensures that a function does not have any 'jumps' or breaks over its domain, which in closed intervals guarantees the attainment of both extreme values. In contrast, discontinuous functions can be manipulated to have specific properties by introducing breaks or removable discontinuities, such as having an absolute maximum without an absolute minimum or vice versa. This concept is essential for understanding how alterations in continuity affect a function’s extrema.

Piecewise Defined Functions

Piecewise defined functions allow different rules for different parts of the domain and can be designed to have customized behaviors, including intentional discontinuities. They are a practical tool for constructing examples with particular properties, such as exhibiting an absolute maximum without an absolute minimum or maintaining extrema despite discontinuities. This flexibility makes them a valuable concept when designing or analyzing functions with atypical behavior.

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