Let $I = [t_{0}, t_{1}]$ be a closed interval with $t_{0} < t_{1}$ and let $M$ be a compact real analytic $n$-dimensional manifold without boundary. Furthermore, let $f:I \times M \rightarrow \mathbb{R}$ be real analytic, non-constat function such that $f_{t_{0}} = f(t_{0}, \cdot), f_{t_{1}} = f(t_{1}, \cdot) : M \rightarrow \mathbb{R}$ are morse functions (in the usual morse theory sense, i.e. non-degenerate hessian at all critical points).
Question: Is it possible to deduce that for almost all $t \in I$ the function $ f_{t} = f(t, \cdot) : M \rightarrow \mathbb{R}$ is a morse function?
Is this true? If its not, then one can add one more additional assumption, namely that for all $t \in I$ the function $ f_{t}$ is non-constant. How about under this additional condition ? Can the Question be true? If this is not true at all, then under what other additional assumptions might this be true. Also, any references are welcome.
There is the Theorem saying that every smooth function on a closed manifold may be approximated by a morse function. But I do not think that this theorem applies in this case.
Best, Bene
