Section 2.5: Continuity lim f (x) = f (a).

Section 2.5: Continuity Definition: A function f is continuous at x = a if lim f (x) = f (a).

x→a

In order for f to be continuous at x = a, the following conditions must hold: 1. f (a) is defined 2. lim f (x) exists x→a

3. lim f (x) = f (a). x→a

If f is not continuous at x = a, we say f is discontinuous at x = a or has a discontinuity at x = a. The three types of discontinuities are holes, jumps, and vertical asymptotes.

Example: Explain why each function is discontinuous at the given point. (a) f (x) =

1 ,x=2 (x − 2)2

 (b) f (x) =

1 − x if x ≤ 2 x2 − 2x if x > 2.

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  x2 − 2x − 8 if x 6= 4 (c) f (x) = x − 4  3 if x = 4.

Example: Find the points at which f is discontinuous   2x + 1 if x ≤ −1 3x if −1 < x < 1 f (x) =  2x − 1 if x ≥ 1.

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Example: Find the constant c that makes g continuous on (−∞, ∞)  g(x) =

x2 − c2 if x < 4 cx + 20 if x ≥ 4.

Example: Find the values of a and b that make   2x ax2 + b f (x) =  4x

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f continuous on R. if x < 1 if 1 ≤ x ≤ 2 if x > 2.

Definition: A function f is continuous from the right at x = a if lim f (x) = f (a).

x→a+

Similarly, f is continuous from the left at x = a if lim f (x) = f (a).

x→a−

Example: The graph of a function f is given below. State the numbers at which f is discontinuous. For each point of discontinuity, state whether f is continuous from the right, left, or neither.

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Definition: A function f is said to have a removable discontinuity at x = a if lim f (x) x→a

exists, but is not equal to f (a). The discontinuity can be removed by redefining f (a) so that lim f (x) = f (a). x→a

Example: Which of the following functions f has a removable discontinuity at x = a? If the discontinuity is removable, find a function g that agrees with f for x 6= a and is continuous on R. (a) f (x) =

x2 + 2x − 8 ,a=2 x−2

(b) f (x) =

x−7 ,a=7 |x − 7|

Theorem: (Intermediate Value Theorem) If f is continuous on the closed interval [a, b] and N is any number strictly between f (a) and f (b), then there exists a number c in (a, b) such that f (c) = N .

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Example: Show that there is a root of the equation 4x3 − 6x2 + 3x − 2 on the interval (1, 2).

Example: If f (x) = x3 − x2 + x, show that there is a number c such that f (c) = 10.

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