The propagation velocity.

  A property related to the smoothness of the solutions is that the heat equation allows an infinite speed of propagation. At any time greater than zero, the temperature at the right end will be greater than zero. It will be extremely small at small times, but never exactly zero. This implies that extremely small amounts of heat from the left half of the bar reach the right end immediately, which is only possible if it can propagate with infinite speed.

You could ignore extremely small amounts of heat and only look at the typical thickness of the layer in the right half in which most of the heat ends up. But even then, the thickness of this layer is proportional to $\sqrt t$, and taking the derivative, the velocity of expansion of the layer is initially infinite. Physically, of course, the velocity of propagation must be finite, so some of the physics has been lost in deriving the heat equation.