Consider a smooth manifold $N$ of finite dimension.
Initial submanifolds of $N$ can be characterized in at least two ways.
(1) First, an initial submanifold of $M$ is an injective immersion $\iota: M \longrightarrow N$ with the property that whenver $f: X \longrightarrow M$ is an arbitrary function $f$ is smooth if (and only if) $\iota f$ is.
(2) Secondly, for $A \subseteq N$ and $x \in X$ denote by $C_x(A)$ the smooth path connected component of $x$ in $A$, i.e. the set of all $y$ in $A$ that can be connected to $x$ via a smooth path that lies in $A$. Now given a subset $M$ of $N$ with the property that for each $x \in M$ there exists a chart $(u,U)$ of $N$ at $x$ such that $u(C_x(M \cap U)) = u(U) \cap (\mathbb{R}^m \times 0)$ (for some fixed $m \in \mathbb{N}$) there is a unique manifold structure (of dimension $m$) on $M$ such that the inclusion $M \longrightarrow N$ becomes an injective immersion and it then has property (1).
A proof of (1) $\Leftrightarrow$ (2)can be found for example in http://www.mat.univie.ac.at/~michor/kmsbookh.pdf, starting on page 13.
From the proof it becomes clear that only a (seemingly) weaker version of (1) is needed to show (2), namely (apart from $\iota$ being an injective immersion) that if $f: X \longrightarrow M$ is arbitrary and $\iota f$ is smooth then $f$ is continuous. Let's call this property (1').
Since clearly also (1) $\Rightarrow$ (1') we have (1) $\Leftrightarrow$ (1').
My first question is now the following: Can initial submanifolds $\iota: M \longrightarrow N$ be characterized amongst injective immersions also by the following property? If $f: X\longrightarrow M$ is an arbitrary function and $\iota f$ is continuous then also $f$ is continuous. (1'')
Clearly (1'') $\Rightarrow$ (1') so only the converse is in question.
Also, if we define $C^0_x(A)$ analogously to $C_x(A)$ from (2) but with continuous paths instead of smooth ones then it is fairly straightforward to see (by slightly adapting the proof from the above link) that (1'') is equivalent to (2''), where (2'') is property (2) with $C_x$ replaced by $C^0_x$.
This brings me to my second question (which, if answered positively, would also answer the first one): For an inital manifold $\iota: M \longrightarrow N$ and $x \in \iota(M)$, is $C_x(\iota(M)) = C^0_x(\iota(M))$?
My intuition of what an initial manifold can typically look like would answer both questions with yes.