In the present paper the authors give an example of a functional \[ F(u)=\int _0^1 f(u(t),u^\prime (t))dt \] defined on the Sobolev space \(W^{1,1}((0,1),\mathbb{R}^2)\) for which the \(L^1\)-lower semicontinuity doesn't hold. This example shows that no reasonable extension of the functional \(F\) to the space BV has a minimizer. The counterexample is construct by considering the function \[ f(\theta,\bar\theta)=f(\xi,\eta,\bar\xi,\bar\eta)= \begin{cases} |\bar\xi -1|+|\bar\eta|+\frac{1}{|\eta|}+|\xi|+|\eta|, & \eta\neq 0 \\ |\bar\xi -1|+|\bar\eta|+|\xi|+|\eta|, & \eta=0. \end{cases} \] The function \(f\) is lower semicontinuous and convex in the \(\bar\theta\) variable; these properties imply a coercivity of the functional \(F\), but this is not enough to garantee the lower semicontinuity with respect to the \(L^1\)-topology.