Does $\int_{a}^{b}f(\phi(x))\phi'(x) \ dx=\int_{\phi(a)}^{\phi(b)}f(t) \ dt$ still hold if $f$ is continuous almost everywhere?

Question

Equip $[0,1]$ with the Borel sigma algebra and lebesgue measure. Suppose that $\phi\colon[0,1]\to[0,1]$ is continuously differentiable. Does the change of variables formula still hold when the integrand $f\colon[0,1]\to\mathbb{R}$ is merely continuous almost everywhere (instead of continuous)? In other words, for $0\leq a\leq b\leq1$, do we still have $$\int_{a}^{b}f(\phi(x))\phi'(x) \ dx=\int_{\phi(a)}^{\phi(b)}f(t) \ dt?$$