Continuity and Differentiability with Parameters
- Ken Schwartz
How variable parameters affect continuity and differentiability of piecewise funcitons.
What makes a function continuous is its "connectedness". An intuitive approach is to see if you can trace the graph without lifting your pencil. Discontinuities occur because of vertical or horizontal gaps (jumps), vertical asymptotes, or "holes". We formalize this by saying that a function is continuous at a point if (1) the limit exists at and (2) the function value equals this limit. When we have a piecewise function , and the value of one or more pieces of depends not only on but on a parameter , we can try to determine one or more values for that makes continuous. For example, suppose for and for . When , the two "pieces" line up, making continuous, but if is changed, the right-hand piece will shift up or down ("breaking" the graph), and is no longer continuous at . To find the value of that ensures continuity, we can begin by setting the left and right piece function definitions equal to each other, then solving for . This ensures that the two one-sided limits are equal and thus that the limit exists. We also check that this value of gives us a defined value for , equal to the limit. If we are not successful finding such a value of , then no value of exists that will make continuous at . In the app, first enter the two function "piece" definitions in the "f1=" and "f2=" boxes. These definitions can include a parameter "". You can change , the value of where the function changes definition, by sliding the red dot along the -axis. You can also determine whether is defined at for either piece or for neither piece by setting the vertical slider to the right of the boxes. (Look at the function definition text below the boxes to see the effect). Then, you can use the slider below the function input boxes to change the value of , and observe the effect on the graph. Differentiability of at works on an analogous concept, expect that the rules are (1) must be continuous, and (2) the derivatives (slopes) of the two pieces must match at , so that the graphs of the two pieces flow smoothly together The app will give you the analysis of at with values for the left and right limits, the function value, the left and right derivatives, and the conclusion whether is continuous and/or differentiable at .