continuous dependence on the data
Every real measurement is a little bit wrong. The initial temperature you feed into a model is only known to a few decimal places; the boundary values drift; the source term is estimated. So a model is only trustworthy if a small error in those inputs produces only a small error in the output. Continuous dependence on the data is the precise promise that this holds: nudge the initial or boundary data a little, and the solution moves only a little. It is the third of Hadamard's three requirements for well-posedness, and in practice it is the one that decides whether a model can be used at all.
Concretely, you measure 'how far apart' two sets of data are, and 'how far apart' the resulting solutions are, using norms — a way of assigning a single non-negative size to a function. Continuous dependence says: if the data of problem 1 and problem 2 differ by a small amount in the data-norm, then the solutions differ by a correspondingly small amount in the solution-norm. The sharpest form is a stability estimate, an inequality of the shape (size of the difference in solutions) <= C times (size of the difference in data), with C a fixed constant not depending on the particular data. Such an estimate is typically proved with an energy method: multiply the equation governing the difference of two solutions by that difference, integrate, and integrate by parts to bound its energy by the data difference. The same one inequality delivers both uniqueness (zero data difference forces zero solution difference) and stability.
This is exactly the property that fails, spectacularly, in an ill-posed problem. In Hadamard's Cauchy problem for Laplace's equation the data shrinks to zero while the solution blows up, so no constant C can hold the inequality — continuous dependence is destroyed. The backward heat equation fails the same way. The presence or absence of continuous dependence is therefore the practical fault line between equations you can compute with and equations you cannot, and it is the reason numerical analysts care so much about stability: a scheme that loses continuous dependence will amplify rounding error until the computed answer is garbage.
For the heat IBVP, let u and v solve the same equation and boundary conditions with initial data f and g. Multiplying the equation for w = u - v by w and integrating gives d/dt of the integral of (1/2) w^2 <= 0, so the integral of w(x,t)^2 never exceeds the integral of (f - g)^2. A small change in the initial data stays small for all time — continuous dependence, with constant C = 1.
An energy estimate bounds the gap between two solutions by the gap between their data — continuous dependence made quantitative.
Continuous dependence is always stated with respect to specific norms; a problem can depend continuously on the data in one norm and not in a stronger one, so 'stable' is meaningful only once you say stable in what measure of size.