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A family given by smooth equations can have individually smooth fibers of different dimensions. For example, letFor $t\ne 0$, this is a circle and a one-dimensional smooth manifold; at $t=0$, it is a single point, a zero-dimensional smooth manifold.

Take $p_t=(t,0)$ and $v_t=(0,1)$ for $t\ne 0$. Differentiating the defining equation in the spatial variables gives $2x{v}_x+2y{v}_y=0$, so $v_t\in T_{p_t}{X}_t$. Yet $p_t\to (0,0)$ and $v_t\to (0,1)$, while $T_{(0,0)}{X}_0=\{0\}$. Thus convergent tangent vectors need not remain tangent to the limiting fiber.

This is not a smooth family in the stronger submersion sense. The total equation defines a cone singular at $(t,x,y)=(0,0,0)$. A submersion between fixed-dimensional smooth manifolds has constant fiber dimension by the submersion theorem. The example distinguishes fiberwise smoothness and smooth dependence of equations from that stronger regular-family hypothesis.