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Intermediate value theorem

Level FundamentalDifficulty ★★★★★Theorem⌖ Open in the map

What is it?

A continuous function on $[a, b]$ takes every value between $f(a)$ and $f(b)$ . Apply Bolzano to $f(x) - y$ .

Statement

$f \in C[a,b]$ and $y$ between $f(a)$ and $f(b)$ $\implies \exists c \in [a,b]:\ f(c) = y$ .

Where it shows up in computing

Inverse kinematics★★★★★indirectRobotics and control
Existence arguments of the type "some joint angle reaches this position" are intermediate-value arguments.

Where is it used?

Computing topics reachable from here, through the chain of ideas that leads to them:

⚙ Robotics and control

Inverse kinematics★★★★★