1. **State the problem:** We want to find the value of $a$ such that the piecewise function $$f(x) = \begin{cases} x^2 & \text{if } x < 1 \\ ax + 1 & \text{if } x \geq 1 \end{cases}$$ is continuous at $x = 1$. 2. **Recall the continuity condition:** A function is continuous at $x=1$ if $$\lim_{x \to 1^-} f(x) = f(1) = \lim_{x \to 1^+} f(x).$$ 3. **Calculate the left-hand limit:** For $x < 1$, $f(x) = x^2$, so $$\lim_{x \to 1^-} f(x) = 1^2 = 1.$$ 4. **Calculate the right-hand limit and function value at 1:** For $x \geq 1$, $f(x) = ax + 1$, so $$f(1) = a \cdot 1 + 1 = a + 1,$$ which is also the right-hand limit: $$\lim_{x \to 1^+} f(x) = a + 1.$$ 5. **Set the limits equal for continuity:** $$1 = a + 1.$$ 6. **Solve for $a$:** $$a = 0.$$ **Final answer:** The function is continuous at $x=1$ if $a = 0$. **Answer choice:** d. 0