Mathematics

The Intermediate Value Theorem guarantees that $f$ hits every value between $f(a)$ and $f(b)$ only when two specific properties hold. First, $f$ must be continuous on the closed interval $[a,b]$—no jumps or breaks allowed. Second, you need two endpoints with different function values, meaning $f(a) \neq f(b)$, so there's actually a range of values to capture. Without continuity, $f$ can "jump over" intermediate values; without distinct endpoints, there's no interval of values to guarantee.

The Intermediate Value Theorem guarantees that $f$ hits every value between $f(a)$ and $f(b)$ only when two specific properties hold. First, $f$ must be continuous on the closed interval $[a,b]$—no jumps or breaks allowed. Second, you need two endpoints with different function values, meaning $f(a) \neq f(b)$, so there's actually a range of values to capture. Without continuity, $f$ can "jump over" intermediate values; without distinct endpoints, there's no interval of values to guarantee.