Integration over level sets

Question

Let $g:R^n \rightarrow R$ be smooth function and let $d:R^n \rightarrow R$ be the signed distance function from the boundary of some open set relatively compact $E\subset R^n$ with smooth boundary so $d$ is smooth in a tubular neighborhood $U=\{x\in R^n: dist(x,\partial E)<b\}$ as well. Now define the function $f:(-b,b)\rightarrow R$ to be $$f(a)=\int_{\{d=a\}} gdH^{n-1}.$$ How to show that $f$ is continuous?