Lebl-Basic-Analysis-I · Problem 3.3.4

The intermediate value property doesn't require continuity everywhere—it only requires that you can't "jump over" a target value. The key insight is to split the interval at $x = 0$ and handle it carefully: on any subinterval away from zero, $f$ is continuous and IVP holds automatically; when zero lies between $a$ and $b$, $\sin(1/x)$ oscillates wildly near zero but stays bounded in $[-1, 1]$, so you can find the target value either in $(a, 0)$ or $(0, b)$ by continuity, or argue directly using the oscillation's density.

The intermediate value property doesn't require continuity everywhere—it only requires that you can't "jump over" a target value. The key insight is to split the interval at $x = 0$ and handle it carefully: on any subinterval away from zero, $f$ is continuous and IVP holds automatically; when zero lies between $a$ and $b$, $\sin(1/x)$ oscillates wildly near zero but stays bounded in $[-1, 1]$, so you can find the target value either in $(a, 0)$ or $(0, b)$ by continuity, or argue directly using the oscillation's density.