1. **State the problem:** We need to determine if the piecewise function $$f(x) = \begin{cases} 0 & \text{if } x < 0 \\ x & \text{if } x \geq 0 \end{cases}$$ is continuous. 2. **Recall the definition of continuity:** A function $$f$$ is continuous at a point $$a$$ if three conditions hold: 1. $$f(a)$$ is defined. 2. The limit $$\lim_{x \to a} f(x)$$ exists. 3. $$\lim_{x \to a} f(x) = f(a)$$. 3. **Check continuity for all $$x \neq 0$$:** - For $$x < 0$$, $$f(x) = 0$$ which is constant and continuous. - For $$x > 0$$, $$f(x) = x$$ which is a polynomial and continuous. 4. **Check continuity at $$x = 0$$:** - $$f(0) = 0$$ (from the second piece since $$0 \geq 0$$). - Left-hand limit: $$\lim_{x \to 0^-} f(x) = 0$$ (since $$f(x) = 0$$ for $$x < 0$$). - Right-hand limit: $$\lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} x = 0$$. - Since left-hand limit, right-hand limit, and $$f(0)$$ are all equal to 0, the function is continuous at $$x=0$$. 5. **Conclusion:** The function $$f(x)$$ is continuous for all real numbers. This is because the function pieces meet at $$x=0$$ without any jump or gap, satisfying the continuity conditions.