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
For the sake of simplicity, we only sketch a proof for and and omit a proof for the fact that and have the same concave modulus of continuity .
Proof. For each , we set
If we only need an extension for , the function constructed above is the desired function since, as we shall see later on . However, if we further want to pursue the two identities and , we need another step.
We shall prove that our (really) extension is
Why ? To see this, first, if , then we obtain
for all ; hence in . In particular, in since we can select as thanks to . From this, we know that for any . Also by definition, for all ; hence as claimed.
Next, for all and for all , we clearly have
Thus, we first have and ; hence . To see the equality, we only need to show that . Once we have this, from the estimate and by taking the infimum all over , we would have . However, this is trivial since in making .
As indicated in Real Analysis by Emmanuele DiBenedetto, Kirzbraun first proved the theorem when is of linear. The general modulus of continuity was proved by Pucci.