Discuss the continuity of the function $f$ given by $$f(x)= \begin{cases}x, & \text { if } x \geq 0 \\ x^{2}, & \text { if } x<0\end{cases}$$

Step 1: Analyze continuity for $x > 0$

For all values of $x > 0$, the function is defined as $f(x) = x$. This is a polynomial function, and all polynomial functions are continuous everywhere in their domain. Therefore, $f(x)$ is continuous for all $x > 0$.

Step 2: Analyze continuity for $x < 0$

For all values of $x < 0$, the function is defined as $f(x) = x^2$. This is also a polynomial function, which means it is continuous everywhere in its domain. Therefore, $f(x)$ is continuous for all $x < 0$.

Step 3: Analyze continuity at $x = 0$

To check continuity at $x=0$, we need to evaluate the left-hand limit, the right-hand limit, and the function value at $x=0$. The left-hand limit is $\lim_{x \to 0^-} x^2 = 0$. The right-hand limit is $\lim_{x \to 0^+} x = 0$. The function value at $x=0$ is $f(0) = 0$. Since all three values are equal, the function is continuous at $x=0$.

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