Step 1: Recall a couple of facts about continuity.
Sums of continuous functions stay continuous, and the absolute value of a continuous function is also continuous. We use these to clear the options quickly.
Step 2: Check $f(x) = |x| + |x+1| + |x-2|$.
Each absolute value is continuous, and a sum of continuous functions is continuous. So this one is continuous everywhere (it only has sharp corners, not breaks).
Step 3: Check $f(x) = |\cos x|$.
Cosine is continuous, and taking its absolute value keeps it continuous. No breaks.
Step 4: Check $f(x) = x^3 + |x|$.
$x^3$ is continuous and $|x|$ is continuous, so their sum is continuous everywhere.
Step 5: Check $f(x) = [x]$, the greatest integer function.
This function jumps by $1$ each time $x$ crosses an integer. At every integer it suddenly leaps, so it is discontinuous there.
Step 6: Pick the odd one out.
Only the greatest integer function has actual breaks, so it is the one not continuous on $\mathbb{R}$. \[ \boxed{f(x) = [x]} \]