1. **Problem Statement:** We analyze the continuity of each function based on its real-world context. 2. **Continuity Definition:** A function is continuous if there are no sudden jumps, breaks, or holes in its graph. Mathematically, for a function $f(x)$ to be continuous at $x=a$, the limit $\lim_{x \to a} f(x)$ must exist and equal $f(a)$. 3. **(a) Temperature at a specific location as a function of time:** - Temperature changes smoothly over time without sudden jumps. - Therefore, this function is generally continuous. 4. **(b) Temperature at a specific time as a function of distance due west from New York City:** - Temperature varies gradually with distance. - No abrupt changes expected in temperature over small distances. - Hence, this function is continuous. 5. **(c) Altitude above sea level as a function of distance due west from New York City:** - Altitude can change abruptly (e.g., cliffs, hills). - This can cause sudden jumps or discontinuities. - So, this function may be discontinuous at some points. 6. **(d) Cost of a taxi ride as a function of distance traveled:** - Taxi fares often have fixed base fees plus per-distance charges. - Cost function can have jumps at fare thresholds (e.g., after certain distances). - Thus, this function is generally discontinuous at those points. 7. **(e) Current in the circuit for the lights in a room as a function of time:** - Electrical current changes smoothly unless switched on/off abruptly. - If lights are switched instantly, current has jumps. - Otherwise, current is continuous. **Final summary:** - (a) Continuous - (b) Continuous - (c) Possibly discontinuous - (d) Discontinuous at fare thresholds - (e) Continuous except at switching times