Let be a continuous function defined on a set with values in and with modulus of continuity

Obviously, the function is nonnegative and nondecreasing in .

Our first assumption is that is bounded from above in by some increasing, affine function; that is to say there exists some such that

.

Associated with having the above first assumption is the concave modulus of continuity of , i.e. some smallest concave function lies above . Such the function can be easily constructed using the following

As can be easily seen, once can be bounded from above by some affine function, the concave modulus of continuity of exists and is well-defined.

By definition and the monotonicity of , we obtain

In this note, we prove the following extension theorem.

Theorem(Kirzbraun-Pucci). Let be a real-valued, uniformly continuous function on a set with modulus of continuity satisfying the first assumption. There exists a continuous function defined on that coincides with on . Moreover, and have the same concave modulus of continuity and