Let $f:X\times [0,1] \to Y$ and let $X$ be compact. There exist points $0= t_0 < t_1 < · · · < t_k = 1$ such that $F(a, t_{i-1}) $,$F(a, t_{i})\in U$ Let $f:X\times [0,1] \rightarrow Y$ be a continuous function, let $X$ and $Y$ be compact Hausdporff, and let  $\mathcal{U}$ be an open cover of $Y$.  If $a\in X$, then there exist points  $0= t_0 &lt; t_1 &lt; · · · &lt; t_k = 1$ and an open subset $U\in \mathcal{U} $ of $Y$  such that  $F(a, t_{i-1}) $,$F(a, t_{i})\in U$ for each $i$. 

I need this result to prove a theorem about simplicial approximations of homotopic maps. Here it is:
I understand your problem to be:  

compact X, f in C(Xx[0,1],Y), C open cover Y, a in X
implies some t0,.. t_k in [0,1], U in C with t0 = 0, t_k = 1
and for j = 1,.. k, f(a,t_(j-1)), f(a,t_j) in U.  

Note that implies for j = 0,1,.. k, f(a,t_j) in U.
A counter example is X = {a}, Y = [0,1], C = { [0,1), (0,1] }  

Could there be a quantification problem?
Is this a correct statement of your problem?  

compact X, f in C(Xx[0,1],Y), C open cover Y, a in X
implies some t0,.. t_k in [0,1], with t0 = 0, t_k = 1 and
for j = 1,.. k, some U_j in C with f(a,t_(j-1)), f(a,t_j) in U.
I understand your problem to be:  

compact X, f in C(Xx[0,1],Y), C open cover Y, a in X
implies some t0,.. t_k in [0,1], U in C with t0 = 0, t_k = 1
and for j = 1,.. k, f(a,t_(j-1)), f(a,t_j) in U.  

Note that implies for j = 0,1,.. k, f(a,t_j) in U.
A counter example is X = {a}, Y = [0,1], C = { [0,1), (0,1] }  

Could there be a quantification problem?
Is this a correct statement of your problem?  

compact X, f in C(Xx[0,1],Y), C open cover Y, a in X
implies some t0,.. t_k in [0,1], with t0 = 0, t_k = 1 and
for j = 1,.. k, some U_j in C with f(a,t_(j-1)), f(a,t_j) in U.