Definition: A function f is continuous at the point x = c if the following 3 criteria are met: A. lim f(x) exists (f has a limit as x c); x-C B. f(c) exists (c lies in the domain of f); C. lim f(x) = f(c) (the limit equals the function value). A function is continuous on the open interval (a, b) if it is continuous at each point on the interval. 1. Create a graph of a continuous function. Pick an x-value, label it "c" on the graph. 2. Create a non-example by graphing a discontinuous function. Pick an x-value, label it "c" where it is discontinuous. 3. Use the definition of continuity to explain why #1 is continuous and #2 is discontinuous. 4. Previously you learned a limit can exist from the left side or right side, creating a one-sided limit. It is also possible to consider a point being continuous from the left side or the right side. Create a graph that is discontinuous according to the definition, but would have a point "e" that is continuous from the left. Create a second graph that is discontinuous according to the definition, but would have a point "c" that is continuous from the right. Continuous from the Left Continuous from the Right 5. What are the intervals of continuity for your two examples? 6. Reflections: Considering non-examples is an important way to explore math concepts. How did the use of your discontinuous functions help you better understand the three criteria required to make a function continuous?