This article introduces the novel notion of dimension preserving approximation for continuous functions defined on [0, 1] and initiates the study of it. Restrictions and extensions of continuous functions in regards to fractal dimensions are also investigated.

We introduce the novel concept of a non-stationary iterated function system by considering a countable sequence of distinct set-valued maps ${\left\{{\mathcal{F}}_k\right\}}_{k\in \mathbb{N}}$ where each ${\mathcal{F}}_k$ maps $\mathcal{H}\left(X\right)\to \mathcal{H}\left(X\right)$ and arises from an iterated function system. Employing the recently-developed theory of non-stationary versions of fixed points and the concept of forward and backward trajectories, we present new classes of fractal functions exhibiting different local and global behavior and extend fractal interpolation to this new, more flexible setting.