The continuity of a function is related to the conceptual notion of a function not having any breaks or sudden jumps as the independent variable takes on all values over some interval. There are no ra...The continuity of a function is related to the conceptual notion of a function not having any breaks or sudden jumps as the independent variable takes on all values over some interval. There are no range gaps, or sudden jumps over the domain of interest.
In order to show the statement is false, all you need is one counterexample where every intermediate value is hit and the function is discontinuous.A counterexample to an if then statement is when the...In order to show the statement is false, all you need is one counterexample where every intermediate value is hit and the function is discontinuous.A counterexample to an if then statement is when the hypothesis (the if part of the sentence) is true, but the conclusion (the then part of the statement) is not true. The converse of the Extreme Value Theorem is: If there is at least one maximum and one minimum in the closed interval [a,b] then the function is continuous on [a,b].
A one sided limit is exactly what you might expect; the limit of a function as it approaches a specific x value from either the right side or the left side. One sided limits help to deal with the issu...A one sided limit is exactly what you might expect; the limit of a function as it approaches a specific x value from either the right side or the left side. One sided limits help to deal with the issue of a jump discontinuity and the two sides not matching.
