Examples: Construct examples of the thing(s) described in...

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading. (a) The graph of a function with $f(4)=2$ that has a removable discontinuity at $x=4$ (b) The graph of a function that is continuous on its domain but not continuous at $x=0$. (c) The graph of a function that is continuous on (0,2] and (2,3) but not on (0,3) .

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.
(a) The graph of a function with $f(4)=2$ that has a removable discontinuity at $x=4$
(b) The graph of a function that is continuous on its domain but not continuous at $x=0$.
(c) The graph of a function that is continuous on (0,2] and (2,3) but not on (0,3) .

Key Concepts

Continuity

Continuity refers to the property of a function where the limit as x approaches any point within its domain equals the function's value at that point. This concept is central in calculus and analysis, ensuring there are no abrupt changes in the graph of a function over intervals that belong to its domain.

Domain of a Function

The domain of a function is the complete set of input values for which the function is defined. In discussions of continuity and discontinuity, it is essential to consider the domain because a function can be continuous on its domain even if there are points (such as isolated points) where it is not defined or extended.

Removable Discontinuity

A removable discontinuity occurs at a point where the limit of a function exists, but the function is either not defined at that point or is defined to have a different value. Such discontinuities are deemed 'removable' because the function can be redefined appropriately at that point to restore continuity.

Piecewise Defined Functions

Piecewise defined functions are constructed using different expressions for different subdomains of the input variable. This technique allows one to create functions that exhibit varying properties, such as being continuous on one interval and discontinuous on another, which is particularly useful in constructing examples with specific discontinuous behaviors.

Jump Discontinuity

A jump discontinuity takes place when the left-hand and right-hand limits at a point exist but are not equal to each other, resulting in a sudden 'jump' in the function's value. This type of discontinuity is common in piecewise functions and is used to illustrate cases where a function is continuous on subintervals but not on an interval spanning across the point of discontinuity.

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