1. **Problem statement:** Sketch the graph of the piecewise function $$f(x) = \begin{cases} x-1 & \text{if } x<3 \\ 1 & \text{if } x\geq 3 \end{cases}$$ and classify the discontinuity at $x=3$. 2. **Formula and rules:** A function is continuous at $x=3$ if $$\lim_{x \to 3^-} f(x) = \lim_{x \to 3^+} f(x) = f(3).$$ If these limits do not match or do not equal $f(3)$, the function is discontinuous. 3. **Calculate left-hand limit:** For $x<3$, $f(x) = x-1$, so $$\lim_{x \to 3^-} f(x) = 3 - 1 = 2.$$ 4. **Calculate right-hand limit:** For $x \geq 3$, $f(x) = 1$, so $$\lim_{x \to 3^+} f(x) = 1.$$ 5. **Evaluate function at $x=3$:** $$f(3) = 1.$$ 6. **Compare limits and function value:** Left limit $= 2$, right limit $= 1$, and $f(3) = 1$. Since left and right limits are not equal, the limit at $x=3$ does not exist. 7. **Conclusion on discontinuity:** Because the left and right limits differ, there is a **jump discontinuity** at $x=3$. 8. **Graph sketch description:** - For $x<3$, the graph is a line with slope 1 and intercept $-1$. - At $x=3$, the function jumps down from the left limit value 2 to the value 1. - For $x \geq 3$, the graph is a constant line at $y=1$. This completes the solution for problem 1.