Ch. 2 – Limits and Continuity

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Presentation transcript:

Ch. 2 – Limits and Continuity

Therefore, f(x) is discontinuous at x = 1 and x = 5 over [0, 5]. A function f(x) is continuous if for all values of c on a specified interval. Basically, the graph can’t have any holes, asymptotes, or breaks! For an interval [a, b], the endpoints are defined as continuous if Ex: Find the points of discontinuity of the function graphed below over [0, 5]. Therefore, f(x) is discontinuous at x = 1 and x = 5 over [0, 5]. Get used to stating the interval in your answer!

Types of Discontinuities (all at x = 2) Removable Discontinuities Jump Discontinuity Infinite Discontinuity Oscillating Discontinuity

Ex: Find the points of discontinuity of the following functions and state the type of discontinuity. f(x) has an infinite discontinuity at x = 0 over the real numbers. f(x) has a removable discontinuity at x = -1 over the real numbers. f(x) is discontinuous outside its domain; that is, from (-∞,-1) and (1, ∞)

Extended Functions Ex: Write an extended function for f(x) that is continuous for all real values of x. Extended function = piecewise function with f(x) and value for all of the removable discontinuities This function has a hole at x = 1, so let’s fill in the hole… Since f(x) approaches -1 at x = 1, our answer is as follows:

Sketch a possible graph.

Sketch a possible graph.