The Intermediate-Value Theorem
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In our first lesson on continuous functions we said that a continuous function is one that can be drawn on a piece of paper without having to lift the pencil from the paper to draw it. We formally call this the intermediate-value theorem.
| The Intermediate-Value Theorem |
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| If $f(x)$ is continuous on $[a,b]$ and <Sign in to see all the formulas> then for every number <Sign in to see all the formulas> between $f(a)$ and $f(b)$ there is at least one number <Sign in to see all the formulas> in $(a,b)$ such that <Sign in to see all the formulas>. |
While you can find the proof here, it is more useful to keep the following picture in mind:
To really gain an intuition around the Intermediate-Value Theorem it is best to look at what happens when a function is not continuous. In the picture below we see that there are values of <Sign in to see all the formulas> for which there is no <Sign in to see all the formulas>.
