$L^2$ property of samples

Assume a continuous, differentiable function $x(t)$ which is uniformly bounded and $L^2$ i.e.$\int_0^\infty{x^2(s)ds}<\infty$ and its derivative is also uniformly bounded. Do we also have that $\sum_{k=0}^\infty x^2(k)<\infty$? If not, what extra assumptions are needed to ensure this property?

My analysis: Since $x,\dot{x}\in L^{\infty}$ and $x\in L^2$ one has from Barbalat's lemma that $x(t)\rightarrow 0$ and therefore $x(k)\rightarrow 0$.