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# Intermediate Value Theorem, Existence of Solutions

## Continuity on an interval proves existence of values and heights within the interval.

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Intermediate and Extreme Value Theorems

### Vocabulary Language: English

continuity

continuity

Continuity for a point exists when the left and right sided limits match the function evaluated at that point. For a function to be continuous, the function must be continuous at every single point in an unbroken domain.
Continuous

Continuous

Continuity for a point exists when the left and right sided limits match the function evaluated at that point. For a function to be continuous, the function must be continuous at every single point in an unbroken domain.
converse

converse

If a conditional statement is $p \rightarrow q$ (if $p$, then $q$), then the converse is $q \rightarrow p$ (if $q$, then $p$. Note that the converse of a statement is not true just because the original statement is true.
counterexample

counterexample

A counterexample is an example that disproves a conjecture.
extreme value theorem

extreme value theorem

The extreme value theorem states that in every interval $[a,b]$ where a function is continuous there is at least one maximum and one minimum. In other words, it must have at least two extreme values.
intermediate value theorem

intermediate value theorem

The intermediate value theorem states that if $f(x)$ is continuous on some interval $[a,b]$ and $n$ is between $f(a)$ and $f(b)$, then there is some $c\in[a,b]$ such that $f(c)=n$.