Suppose $(M,g)$ is a complete Riemannian manifold. $p\in M$ is a fixed point. $d\_{p}(X)$ is the distance function defined by $p$ on M (i.e., $d\_p(x)$=the distance between $p$ and $x$). Let $\epsilon>0$ be an arbitrary positive number. Is there a smooth function $\tilde{d}\_p(x)$ on $M$, such that 
$$ | d\_p(x)-\tilde{d}\_p(x) | < \epsilon$$
$$ |\textrm{grad}(\tilde{d}\_p)(x)|<2$$
for $\forall x \in M$ ?


Functions satisfying the first condition can be constructed easily by partition of unity and the standard technique of mollifiers. However, I can't see how to control the gradient of the approximate function.


I need this result when I'm trying to follow a proof. The existence of such a function is taken for granted in that proof, and is called the "regularization" of the distance function. This question also seems to be interesting by itself. 


I'm not sure if this question is too easy for math overflow. I put it on stack exchange last week but nobody answered it. Could you please help me? Thanks a lot.



Smoothing of the distance function on a Riemannian manifold
You function $d\_p$ is a Lipschitz function (w.r.t. to the Riemannian distance) with the Lipschitz constant $1$ and it is possibly, for every $\varepsilon>0$, to $\varepsilon$-approximate it by a smooth function with the length of the gradient $\le 1 + \varepsilon$. 


The proof of this statement, even if we replace 'Riemannian' by 'Finslerian', can be found in the appendix to my paper <http://arxiv.org/abs/1112.5060>; the proof seems to be wellknown for experts so I did not pretend that the proof is new but tried to explain it in all details. In the Riemannian category, the statement was proved in D. Azagra, J. Ferrera, F. Lopez-Mesas, Y. Rangel, Smooth approximation of Lipschitz functions on Riemannian manifolds, J. Math. Anal. Appl. 326(2007) 1370–1378. I should confess that I found it complicated to read this paper but if you need it as a citation and not in order to understand the proof it could be an option 


