Let \(f\) be a real function of bounded variation on \([a,b]\) . Denote its total variation on that interval by \(\bigvee_{a}^{b}\left( f\right) \) . The authors prove the following inequality \[ \left|\int_{a}^{b}f(t)dt-f(a)(x-a)-f(b)(b-x)\right|\leq \left[ \frac{1}{2} (b-a)+\left|x-\frac{a+b}{2}\right|\right] \bigvee_{a}^{b}\left( f\right) \] for all \(x\in [a,b]\) . The constant \(1/2\) is best possible. Some applications to quadrature formulae and special means are also given.