Continuity and the Intermediate Value Theorem

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Continuity and the Intermediate Value Theorem

Definition of continuity
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The Intermediate Value Theorem
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The Fundamental Theorem of Calculus

Three Different Quantities
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The Intermediate Value Theorem

If a function $f$ is continuous at every point $a$ in an interval $I$ , we'll say that $f$ is continuous on $I$ .

The Intermediate Value Theorem talks about the values that a continuous function has to take:

Theorem: Suppose $f(x)$ is a continuous function on the interval $[a,b]$ with $f(a) \ne f(b)$ . If $N$ is a number between $f(a)$ and $f(b)$ , then there is a point $c$ between $a$ and $b$ such that $f(c)=N$ .

In other words, to go continuously from $f(a)$ to $f(b)$ , you have to pass through $N$ along the way. In this video we consider the theorem graphically and ask: What does it do for us?