2.3 Continuity Grand Canyon, Arizona Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts

Most of the techniques of calculus require that functions be continuous. A function is continuous (in a casual way) if you can draw it in one motion without picking up your pencil… a more thorough definition: A function is continuous at a point if the limit is the same as the function value at that point. The function at the left has discontinuities at x=1 and x=2. f(x) is continuous at x=0 and x=4, because the one-sided limit matches the function value at the endpoint lim f(x) = f(c) = lim f(x) xc- xc+

jump infinite oscillating Essential (Non-Removable!) Discontinuities: Removable Discontinuity: (You could fill the gap and establish continuity by adding or moving a single point.)

To write an extended function that is continuous at x = 1, find the limit of f(x) approaching x = 1… Removing a point discontinuity: … and add a new piece to the original function: the new height is the limit value (note its x-coordinate). The new function is an extended, now-continuous function. has a removable discontinuity at x=1… (Note: The essential discontinuity at x = -1 cannot be removed!)

Removing a discontinuity from a graphic perspective: (Also note the discontinuity at x = -1 that cannot be removed.) ° Note that the original hole in the graph has now been filled in!

Continuous functions can be added, subtracted, multiplied, divided, multiplied by a constant, or composed, and the new function remains continuous. Composites of continuous functions are continuous! examples: Outer function: y = sin( ) Inner function: y = ( ) 2 Outer function: y = abs( ) Inner function: y = cos( )

Intermediate Value Theorem If a function is continuous between a and b, then it takes on every function value between and Because the function is continuous, f reaches every height -value between f(a) and f(b). (Sometimes that height is reached more than once!)

Example 5: Is any real number exactly one less than its cube? Since f is a continuous function, by the IVT, f must take on every function value between -1 and 5. Therefore, there must be at least one solution (value of 0) between 1 and 2. Use your calculator to approximate that solution, via 2 nd TRACE zero… f(1) = -1 … … a negative height and f(2)= 5 … … a positive height x

Graphing calculators can make non-continuous functions appear to be continuous… look out! Graph: MATH 5 int( Note x- resolution. In connected mode, the calculator can connect the dots, and cover up the discontinuities at every integer. NUM

Graphing calculators can make non-continuous functions appear continuous … look out! The open and closed endpoints do not show, but we can see the discontinuities more clearly! If we change the x-resolution to 1, then we get a graph that is closer to the actual step graph. Graph: