Let $E,F,G$ be topological vector areas such that $Fsubset G$ with steady embedding.
By continuity improve I imply the next phenomenon: In some circumstances a steady linear map $f:Erightarrow G$ with $f(E)subset F$ is routinely steady as map $f:Erightarrow F$. That is true e.g. if all concerned areas are Fréchet (after which follows from the closed vary theorem).
I’m involved in nonlinear variations of this phenomenon. One instance for that is if $f$ is a quadratic kind, i.e. $f(x)=b(x,x)$ for a bilinear map $b:Etimes Erightarrow G$. On this case one can use the linear outcome for $b(x,cdot)$ and $b(cdot,y)$ to conclude that $b:Etimes E rightarrow F$ is individually steady in every variable and (at the very least within the Fréchet setting) this suggests joint continuity and thus continuity of $f:Erightarrow F$.
Query. Let $f:Erightarrow G$ be a steady (presumably non-linear) map with $f(E)subset F$. Which situations on $f$ (e.g. development situations, algebraic situations) and on the concerned areas be sure that $f$ is steady as map $f:Erightarrow F$?