Function $f(x,y)$ is continuous increasing on both dimensions. $f(x,y)$ is locally strictly quasiconcave on a non-linear curve $\gamma\subset\mathbb R$ (i.e. $\{(x,y)|f(x,y)=\lambda\}\cap \gamma$ is convex to origin $\forall\lambda$).
Does this implies that $f(x,y)$ must be also quasiconcave on some convex subset of $\mathbb R$?
Strictly qusiconcave means $f(\lambda a+(1-\lambda) b))<\max\{f(a),f(b)\} \ \forall a,b\in\mathbb R^2$, and $\lambda\in(0,1)$
or, alternatively, $C=\{a\in\mathbb R^2|f(a)\geq\lambda\}$ is a strictly convex set $\forall\lambda\in\mathbb R$