What about $t \mapsto (t,t)$ and $t \mapsto (t+1,t+1)$ for $t \in [0,2]$? They have infinite intersection but do not coincide.

On the positive side (following Peter´s answer and Noam´s comment), the set of $t$´s for which there is some $u$ such that $f(t)=g(u)$ is a finite union of points and intervals by the o-minimality of the structure involved.

Two different analytic curves cannot intersect in infinitely many points
I'm trying to solve a two point boundary value problem with a piecewise system of equations like:

dx[t_] := Piecewise[{{0, y[t] &lt;= 0}, {x[t]^(3/4) - x[t], y[t] &gt; 0}}]
dy[t_] := Piecewise[{{0, y[t] &lt;= 0}, {(1 - y[t])/x[t]^(1/4) + y[t], y[t] &gt; 0}}]

NDSolve[{x'[t] == dx[t], y'[t] == dy[t], x[0] == 1, y[10] == 2}, {x, y}, {t, 0, 10}]


which produces the error

 NDSolve::bvdisc: NDSolve is not currently able to solve boundary value problems with discrete variables.


I tried the answers here by entering

funs = ParametricNDSolveValue[{x'[t] == dx[t], y'[t] == dy[t], x[0] == 1, y[10] == yt}, {x, y}, {t, 0, end}, yt]

FindRoot[funs[yt][end], {yt, 2}]


which yields the same error.

Is there any way to solve this system? Thanks for the help.
