Let $f$ be an interval map in a neighborhood of the fixed point $0$ with \(-1<\lambda=f'(0)<0\). Continuity of $f$ is not assumed at points other than the fixed point. It is shown that if either $$f\circ f(x)\geq\lambda^{2}x \text{ or }f\circ f(x)\leq\lambda^{2}x$$ for each $x$ in a neighborhood of $0$, then the Koenigs' sequence \(\{\phi_{k}\}\) defined by $\phi_{k}(x)=\dfrac{f^{k}(x)}{\lambda^{k}}$ converges uniformly to a limit \(\phi\) in a neighborhood of $0$ with \(\phi(0)=0\) and \(\phi'(0)=1\). Two examples are presented, the first of which is a \(C^{1}\) map $f$ with \(f(0)=0\) and \(-1