1. The problem is to analyze the piecewise function $$y=\begin{cases} x & \text{if } x<0 \\ x^2 & \text{if } x\geq 0 \end{cases}$$. 2. This function is defined differently for negative and non-negative values of $x$. 3. For $x<0$, the function is linear: $y=x$. 4. For $x\geq 0$, the function is quadratic: $y=x^2$. 5. To understand the behavior, we check continuity at $x=0$: $$\lim_{x\to 0^-} y = 0$$ $$\lim_{x\to 0^+} y = 0^2 = 0$$ 6. Since both limits and the function value at $x=0$ are equal, the function is continuous at $x=0$. 7. The graph consists of a line with slope 1 for $x<0$ and a parabola opening upwards for $x\geq 0$. 8. The function is differentiable everywhere except possibly at $x=0$; checking derivatives: $$\frac{dy}{dx} = \begin{cases} 1 & x<0 \\ 2x & x>0 \end{cases}$$ At $x=0$, left derivative is 1, right derivative is 0, so not differentiable at $x=0$. Final answer: The function is continuous everywhere but not differentiable at $x=0$.