 Let \(f\) be a continuous real-valued mapping on the closed interval \([0,1]\) such that \(f(0)\cdot f(1) \lt 0\). Suppose also that \(f\) is locally nonzero, in the sense that for each \(x \in [0,1]\) and each \(r \gt 0\), there exists \(y\) such that \(\abs{x - y} \lt r\) and \(f(y) \ne 0\). Then there exists \(x\) such that \(0 \lt x \lt 1\) and \(f(x) = 0\).