Extension of a function from a subset of a manifold to the unit sphere

Question

There is a line in this following paper (page no- 221, the paragraph before the Lemma 2.2)

Otsu, Yukio; Shiohama, Katsuhiro; Yamaguchi, Takao, A new version of differentiable sphere theorem, Invent. Math. 98, No.2, 219-228 (1989). ZBL0688.53016.
saying that the constructed functions $f:M'\rightarrow S^n$ and $f^{-1}:f(M')\rightarrow M$ have natural extensions $M\rightarrow S^n$ and $S^n\rightarrow M$, which are not necessarily continuous... But I can not understand how we can extend this functions. Please help.
Thank you. (This paper can be read from this link: https://eudml.org/doc/143726)