is it ever possible for a sequence of real-valued smooth, non-diffeomorphic functions in $\mathbb{R}$ to jump across a repulsive fixed-point? Let's say we have a real-valued smooth function $f(x)$, not diffeomorphic, ...--prophetes.ai

is it ever possible for a sequence of real-valued smooth, non-diffeomorphic functions in $\mathbb{R}$ to jump across a repulsive fixed-point? Let's say we have a real-valued smooth function $f(x)$, not diffeomorphic, which has an attractive fixed-points at $x=3$ and $x=5$ and a repelling fixed-point at $x=4$ . Is it possible that any sequence of iterated compositions of $f(x)$ could converge to $x=5$ if started from a point $y<3$ ?

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