Problem: Determine whether the function $f(x)=\begin{cases}0 & x<0\\ x & x\ge 0\end{cases}$ is continuous. 1. Recall the definition: a function is continuous at a point $a$ if three conditions hold: (i) $f(a)$ is defined. 2. (ii) $\lim_{x\to a} f(x)$ exists. 3. (iii) $\lim_{x\to a} f(x)=f(a)$. 4. We check continuity on three regions: $x<0$, $x>0$, and at $x=0$. 5. For $x<0$: the function equals $0$ on $(-\infty,0)$, which is a constant function and therefore continuous on that interval. 6. For $x>0$: the function equals $x$ on $(0,\infty)$, which is continuous (identity function) on that interval. 7. At $x=0$ we compute the left-hand and right-hand limits and the value of the function. 8. Left-hand limit: $\lim_{x\to0^-} f(x)=\lim_{x\to0^-} 0 = 0$. 9. Right-hand limit: $\lim_{x\to0^+} f(x)=\lim_{x\to0^+} x = 0$. 10. Value at zero: $f(0)=0$ because the rule for $x\ge 0$ gives $f(0)=0$. 11. Since the left-hand limit, right-hand limit, and $f(0)$ all equal $0$, the limit exists and equals the function value at $0$, so $f$ is continuous at $0$. Conclusion: Therefore $f$ is continuous for all real numbers, i.e., $f$ is continuous on $\mathbb{R}$. Final answer: $f$ is continuous on $\mathbb{R}$.