2. CONTINUITY Mrs. Erickson

3. Continuity lim f(x) = f(c) at every point c in its domain. To be continuous, lim f(x) = lim f(x) = lim f(c) x  c+x  c+ x  c-x  c-

4. Continuity ContinuousRemovable Discontinuities because you can remove the fact that this function is discontinuous. “Fill the hole” and it becomes continuous.

5. Continuity Jump DiscontinuityInfinite DiscontinuityOscillating Discontinuity

6. Continuity Polynomial functions Rational functions Absolute value function Exponential Function Natural log function Trig functions Radical functions Continuous at every point in the domain. Zeros in denominator are points of discontinuity, however they are not in the domain. f(x) = a n x n + a n-1 x n-1 + … + a 1 x + a 0 f(x) =, q(x) ≠ 0 p(x) q(x) y = |x| y = e x y = ln(x) y = sin(x), y = cos(x), y = tan(x), … y = √x

7. The Intermediate Value Theorem f(x) is a continuous function on [a,b]. f(x) takes on every value between f(a) and f(b). There exists a value c, where a<c<b, such that f(c) is in the interval [f(a),f(b)]. If f(x) is discontinuous, then the Intermediate Value Theorem may fail.

8. The End