Openness and monotoneity of induced mappings

It is shown that for locally connected continuum $X$ if the induced mapping $C(f): C(X) \to C(Y)$ is open, then $f$ is monotone. As a corollary it follows that if the continuum $X$ is hereditarily locally connected and $C(f)$ is open, then $f$ is a homeomorphism. An example is given to show that local connectedness is essential in the result.