This is an announcement for the paper "A note on the Cheng-Dong-Zhang Theorem and its applications" by Duanxu Dai. Abstract: In this paper, we first give a short introduction to recent development on the stability of Banach spaces via $\eps$-isometry and then present an application of the Cheng-Dong-Zhang Theorem to the continuous selections of a set valued map via $\eps-$ isometries. Archive classification: math.FA Mathematics Subject Classification: 46B04, 46B20, 54C60 (Primary) 26E25, 46A20, 54C65 (Secondary) Remarks: 7 pages Submitted from: dduanxu@163.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1301.3396 or http://arXiv.org/abs/1301.3396