Let $S$ be a dynamical system on a metric space $X$ with discrete time $\mathbb{N}_0$.
In our script we have a theorem that says one can extend such a system to one, here called $\tilde{S}$, with continuous time $[0,\infty)$ on a larger space $Y$ and $\tilde{S}(1)|_X=S(1)$.
We don't have a proof, only the hint to use $Y=C^0([0,1])$. Can anyone help me with a proof?
You construct an equivalence relation on $X\times [0,1]$ so that $(x,1)$ is equivalent to $(f(x),0)$ where $f=S(1,\cdot)$ is the step-1 map of the dynamical system $S$. Then define $$\tilde S(t,(x,s))=(f^n(x),\alpha)$$ where $s+t=n+\alpha$, $n\in\Bbb N_0$ and $\alpha\in[0,1)$.
Adapt to your notation convention if I guessed the interpretation of your symbols wrong.
Justification: The task description tells that $\tilde S$ acts on a bigger space $\tilde X$ where $X$ is embedded, $\iota:X\to\tilde X$. What we originally know with certainty is that the dynamic on the embedded set should be inherited, $$\tilde S(n,\iota(x))=\iota(S(n,x))=\iota(f^n(x)).$$
What this construction does is to find the most trivial space that satisfies the demands of the task, essentially turning the problem description into a solution.
So a value of $\tilde S(t,\iota(x))$ is needed for $t\in(0,1)$. The easiest construction is to give the pair $(x,t)\in X\times\Bbb R$ as that value. Next one has to ensure continuity at $t=1$, $$ \lim_{t\to 1}\tilde S(t,\iota(x))=\tilde S(1,\iota(x))=\iota(f(x)). $$ This leads directly to the construction of an equivalence relation $(x,1)\sim (f(x),0)$, easily generalized to $(x,t+n)\sim (f^n(x),t)$. Then $\tilde X=(X\times\Bbb R)/\sim$, the topology is inherited from $X\times\Bbb R$, the continuity of $\tilde S$ in all arguments follows from the continuity of $f$.