Suppose $C$ is a collection of mathematical objects (for instance, sets, or functions). Then we say that $C$ is rigid if every $c\in C$ is uniquely determined by less information about $c$ than one would expect.

It should be emphasized that the above ``definition'' does not define a mathematical object. Instead, it describes in what sense the adjective rigid is typically used in mathematics, by mathematicians.

Let us illustrate this by some examples:

1. Harmonic functions on the unit disk are rigid in the sense that they are uniquely determined by their boundary values.
2. By the fundamental theorem of algebra, polynomials in $\sC$ are rigid in the sense that any polynomial is completely determined by its values on any countably infinite set, say $\sN$ , or the unit disk.
3. Linear maps between vector spaces $X,Y$ are rigid in the sense that any is completely determined by its values on any set of basis vectors of $X$ .
4. Mostow's rigidity theorem