4. Let the function be
$$
f(x)=\left(\frac{1}{2}\right)^{x}+\left(\frac{2}{3}\right)^{x}+\left(\frac{5}{6}\right)^{x}(x \in[0,+\infty)) \text {. }
$$

Then the number of integer points on the graph of the function is $\qquad$
4.3.

It is known that the function $f(x)$ is monotonically decreasing on the interval $[0,+\infty)$, and
$$
f(0)=3, f(1)=2, f(3)=1 .
$$

When $x>3$, we have
$$
0<f(x)<f(3)=1 \text {. }
$$

Therefore, the number of integer points on the graph of the function $y=f(x)(x \in[0,+\infty))$ is 3.