The book "The Topology of CW Complexes" mentions the following result in passing, but does not provide a proof. Let $(X, \mathcal{C})$ be a CW complex, $B_k \subset \mathbb{R}^k$ be the origin-centered open unit-$k$-ball, and $\overline{B_k}$ be the corresponding closed $k$-ball. Let $f, g : \overline{B_k} \to X$ both be characteristic functions of the same cell $C \in \mathcal{C}$. Then there exists a homeomorphism $h : \overline{B_k} \to \overline{B_k}$ such that $f = g \circ h$. How can you prove this statement?
For a regular cell, the statement is of course trivial, so I am specifically interested in the non-regular case.
EDIT
In the light of the counter-example given by Eric Wofsey, I may have misunderstood what is said in the book. The direct quotes are as follows ($\dot{E}_n$ is the boundary of a closed cell $E_n$).
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Page 1:
A closed Euclidean $n$-cell $E^n$ is a homeomorphic image of the Euclidean n-cube $I^n$, the cartesian product of $n$ copies of the closed unit interval $I = \{t \in \mathbb{R} : 0 \leq t \leq 1\}$.
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Let $X$ be a set. A cell structure on $X$ is a pair $(X, \Phi)$, where $\Phi$ is a collection of maps of closed Euclidean cells into $X$ satisfying the following conditions.
If $\phi \in \Phi$ and $\phi$ has domain $E_n$, then $\phi$ is injective on $E_n - \dot{E}_n$.
The images $\{\phi(E_n - \dot{E}_n) : \phi \in \Phi\}$ partition $X$, i.e., they are disjoint and have union $X$.
If $\phi \in \Phi$ has domain $E_n$, then $\phi(\dot{E}_n) \subset \bigcup \{\psi(E_k - \dot{E}_k) : \psi \in \Phi \text{ has domain } E_k \text{ and } k \leq n - 1\}$
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We say that two cell structures $(X, \Phi)$ and $(X, \Phi')$ are strictly equivalent if there is a one-to-one correspondence between $\Phi$ and $\Phi'$ such that a characteristic function with domain $E_n$ corresponds to a characteristic function with domain $E_n$, and corresponding functions differ only by a reparametrization of their domain. That is, if $\phi$ and $\phi'$ are corresponding functions of $\Phi$ and $\Phi'$, respectively, then $\phi' = \phi \circ h$, where $h: (E_n, \dot{E}_n) \to (E_n, \dot{E}_n)$ is a homeomorphism of pairs. We leave it to the reader to check that this is an equivalence relation on the collection of cell structures on the set X.
If $(X, \Phi)$ is a cell structure, let $\mathcal{S}_\Phi$ consist of all pairs $(\sigma^n, [\phi])$, where $\sigma^n = \phi(E^n)$ and $[\phi]$ is the strict equivalence class of $\phi \in \Phi$. For convenience, we will denote such a pair by $\sigma^n$ or $\phi(E^n)$ with the class of $\phi$ understood. Note that if $(X, \Phi)$ and $(X, \Phi')$ are strictly equivalent cell structures, then $\mathcal{S}_\Phi = \mathcal{S}_{\Phi'}$.
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A cell complex on a set $X$ or a cellular decomposition of a set $X$ is an equivalence class of cell structures $(X, \Phi)$ under the equivalence relation of strict equivalence.
A cell complex on $X$ will be denoted by a pair $(X, \mathcal{S})$, where $\mathcal{S} = S_{\Phi}$ for some representative cell structure $(X, \Phi)$. The set $\mathcal{S}$ is called the set of (closed) cells of $(X, \mathcal{S})$.
Page 41:
A Hausdorff space $X$ is a CW complex with respect to a family of cells $\mathcal{S}$ provided:
the pair $(X,\mathcal{S})$ is a cell complex such that each cell $\sigma \in \mathcal{S}$ has a continuous characteristic function;
the space $X$ has the weak topology with respect to $\mathcal{S}$;
the cell complex $(X,\mathcal{S})$ is closure finite.
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I'm not sure where my misunderstanding is, but I suspect it has to do with that the strict equivalence is defined between cell-structures and not between characteristic functions.
EDIT 2
Referring to the discussions with Paul Frost below, the confusion arises because the definition in the book is more granular than the usual definition of a CW complex. Therefore, if characteristic functions are not related by a homeomorphism in this definition, then they are not the same CW complex.
In contrast, the usual definition of a CW complex requires only the existence of a characteristic function, which makes it possible to provide non-homeomorphically related characteristic functions for the same CW complex, as provided by the counter-example in the accepted answer.
This is false. If $k>1$, then there exists a map $f:\overline{B_k}\to\overline{B_k}$ which restricts to a homeomorphism $B_k\to B_k$ and maps $\partial B_k$ to itself but which is not injective on $\partial B_k$. So, taking $X=\overline{B_k}$ with a CW-complex structure that has $B_k$ as a $k$-cell, both $f$ and $g=1_{\overline{B_k}}$ are possible characteristic functions for the $k$-cell. Since $f$ is not a homeomorphism, there cannot exist any homeomorphism $h:\overline{B_k}\to\overline{B_k}$ such that $f=g\circ h$.
For an explicit example of such an $f$ for $k=2$, let $S=[0,1]\times[0,1]$ and $T=\{(x,y)\in\mathbb{R}^2:0\leq y\leq x\leq 1\}$ and define $f_0:S\to T$ by $f_0(x,y)=(x,xy)$. Then $f_0$ is a homeomorphism on the interiors of $S$ and $T$ but collapses one of the sides of $S$ to a single point. Composing with homeomorphisms $S,T\cong \overline{B_2}$ we get $f:\overline{B_2}\to\overline{B_2}$ which is a homeomorphism on the interior but collapses an arc on the boundary to a point.