Continuity

Identifying continuity at a point and over an interval, end‐ behavior models, hole(s), oscillating functions, and jump discontinuity in functions. Classifying discontinuities and removing them by factoring and rationalization.

Mapped to AP College Board # LIM-1.C.3, LIM-1.C.4, LIM-2.A, LIM-2.A.1, LIM-2.A.2, LIM-2.B, LIM-2.B.1, LIM-2.B.2, LIM-2.C, LIM-2.C.1, LIM-2.C.2

Because of issues of scale, graphical representations of functions may miss important function behavior. A limit might not exist for some functions at particular values of x. Some ways that the limit might not exist are if the function is unbounded, if the function is oscillating near this value, or if the limit from the left does not equal the limit from the right. Justify conclusions about continuity at a point using the definition. Types of discontinuities include removable discontinuities, jump discontinuities, and discontinuities due to vertical asymptotes. A function f is continuous at x = c provided that f(c) exists, lim f(x) exists for x→c, and lim f(x) = f(c) for x → c. Determine intervals over which a function is continuous. A function is continuous on an interval if the function is continuous at each point in the interval. Polynomial, rational, power, exponential, logarithmic, and trigonometric functions are continuous on all points in their domains. Determine values of x or solve for parameters that make discontinuous functions continuous, if possible. If the limit of a function exists at a discontinuity in its graph, then it is possible to remove the discontinuity by defining or redefining the value of the function at that point, so it equals the value of the limit of the function as x approaches that point. In order for a piecewise-defined function to be continuous at a boundary to the partition of its domain, the value of the expression defining the function on one side of the boundary must equal the value of the expression defining the other side of the boundary, as well as the value of the function at the boundary