Let $f:\mathbb{R}^d\to \mathbb{R}$ be a Lipschitz function on $\mathbb{R}^d$.
The co-area formula states that for all measurable function $\phi$ we have $\int_{\mathbb{R}^d} \phi \|\nabla f\| dx=\int_{\mathbb{R}}dy\int_{f^{-1}(y)} \phi(x) dH_{n-1}(s)ds $ where $H_{n-1}$ is (n-1) Hausdorff measure.
Is there an easy proof of such a result when $f$ is smooth ? Do you get any additional information (for instance continuity with respect to $y$ of $\int_{f^{-1}(y)} \phi(x) dH_{n-1}(s)ds$ if $\phi$ is continuous) ?
I am in fact interested in the case where $f$ is smooth and striclty convex with a unique minimum.