Let's suppose I have a $C^1$ function $f:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$, $(x,\lambda)\mapsto f(x,\lambda)$. Suppose there is a unique solution of the equation $f(x,\lambda_1)=0$, and either zero or more than one solution of the equation $f(x,\lambda_2)=0$, where, say, $\lambda_1<\lambda_2$. I've laboriously proven to myself that at least one of the following must occur.
There exists $\lambda^*\in[\lambda_1,\lambda_2]$ and $x^*\in\mathbb{R}$ such that $f(x^*,\lambda^*)=0$ and $\partial_x f(x^*,\lambda^*)=0$.
The set $\{x\in\mathbb{R} : f(x,\lambda)=0, \lambda\in[\lambda_1,\lambda_2]\}$ is unbounded.
The unsatisfactory thing is that my proof is a good three pages of nit-pick cases, endless use of the implicit function theorem and several awkward sequence constructions. The proof could be shortened by sufficient hand-waving, but I wanted to do it right.
Is there a "nice" proof of this? Any reference would be helpful (I'm not exactly sure what the search terms should be.. :S). Or, is the result incorrect, and something went terribly wrong in my proof?