Let $(E,\pi,B)$ be a locally trivial fibration, with fiber a topological space $F$, $\Phi_i$ and $\Phi_j$ two trivializations over $U_i$ and $U_j$. The transition map from $i$ to $j$ is the homeomorphism $$\Phi_{ji} \colon U_i \cap U_j \times F \to U_i \cap U_j \times F, \quad (x,y) \mapsto (x,\phi_{ji,x}(y)), \quad \phi_{ji,x} \in Homeo(F).$$
Since $pr_2 \circ \Phi_{ji}$ is continuous, from the exponential law $Top(U_i \times F, F) \to Top(U_i \to F^F)$, with the compact-open topology on $F^F$ and its trace on $Homeo(F) \subset F^F$, there is a continuous map $$\phi_{ji} \colon U_i \cap U_j \to Homeo(F), \quad x \mapsto \phi_{ji,x}.$$
In general $Homeo(F)$ is not a topological group because in general the composition is not continuous (it is if $F$ is locally compact) and the inverse map is not continuous (it is if $F$ is compact, or non-compact but locally compact and locally connexe), and **we cannot assume that the map
$$\psi_{ji} \colon U_i \cap U_j \to Homeo(F), \quad x \mapsto \psi_{ji}(x) = (\phi_{ji,x})^{-1}$$ is continuous**. But since $\Phi_{ji}$ is a homeomorphism, its inverse is the continuous map $$\Psi_{ij} \colon U_i \cap U_j \times F \to U_i \cap U_j \times F,\quad (x,y) \mapsto (x,\psi_{ij,x}(y)),\quad \psi_{ij,x} \in Homeo(F),$$
From the exponential law again, we get again a continuous map
$$\psi_{ij} \colon U_i \cap U_j \to Homeo(F), \quad x \mapsto \psi_{ij,x},$$ and since it is obvious that $\psi_{ij,x} = (\phi_{ji,x})^{-1}$, we get that the map $\psi_{ij} \colon x \mapsto \psi_{ij,x} = (\phi_{ji,x})^{-1}$ is continuous.

So can we or can't we say that the map $x \mapsto (\phi_{ji,x})^{-1}$ is continuous in general ?

A similar reasonning can be made for composition : $x \in U_i \cap U_j \cap U_k \mapsto \phi_{ki,x} = \phi_{kj,x} \circ \phi_{ji,x}$ is continuous although the composition in $Homeo(F)$ is continuous only under conditions s.t. $F$ locally compact.

Is there something wrong hidden in what I say ? Or does that mean that the subset of $Homeo(F)$ image of the exponential (i.e. obtained through a continuous map like $\Phi_{ji}$) is in fact a topological group ?