MAT 1234 Calculus I Section 1.8 Continuity

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Presentation transcript:

MAT 1234 Calculus I Section 1.8 Continuity

Preview Continuous at a point Continuous on an open interval How to demonstrate that a function is not continuous Intermediate Value Theorem

Preview Polynomials are “nice” functions when computing limits Let us look into this “nice” property Other “nice” functions

1. Continuous at a Point

Definition A function f is continuous at a point a if

Definition A function f is continuous at a point a if That means…

Example 0 (Section 1.5) x y y=f(x)

Example 0 (Section 1.5) x y y=f(x)

Example 1 Explain why the fun. is discontinuous at x=1 Note that we need to explain this theoretically. The diagram only serves as an aid.

Example 2 Explain why the fun. is discontinuous at x=1

2. Continuous on an Open Interval

Definition A function f is continuous on an open interval if it is continuous at every number of the interval.

What if … If the interval is not open, the definition above breaks down at the end points.

What if … If the interval is not open, the definition above breaks down at the end points. A different (modified) definition is required. We will skip that. Read the text if you are interested.

Common Continuous Functions Polynomials Rational Functions Root Functions Tri. Functions Continuous at every no. in their domains

Combinations of Continuous Functions If f and g is continuous at a, then f+g, f-g, fg, f/g*, cf are also continuous at a. (*g(a)≠0)

Combinations of Continuous Functions If g is continuous at a, and f is continuous at g(a), then the composite function is also continuous at a.

Combinations of Continuous Functions If g is continuous at a, and f is continuous at g(a), then the composite function is also continuous at a. In practice, find the composite function and determine its continuity.

Example 3

Example 4 For what value of c is the function f continuous at ?

Example 4 For what value of c is the function f continuous at ?

Expectations

3. Intermediate Value Theorem

Intermediate Value Theorem Suppose f is continuous on [a,b] with f(a)≠f(b) and N is between f(a) and f(b) Then there is a no. c in (a,b) such that f(c)=N

Intermediate Value Theorem Suppose f is continuous on [a,b] with f(a)≠f(b) and N is between f(a) and f(b) x y b y=f(x) a f(a) f(b) N

Intermediate Value Theorem Applications Use to prove other theorems Use to estimate the roots of equations Find a such that f(a)=0

Intermediate Value Theorem Applications x y b y=f(x) a f(a) f(b) c 0 Suppose f(a) and f(b) are with different signs Then there is a no. c in (a,b) such that f(c)=0

Review: We learned… Continuous at a point Continuous on an open interval Intermediate Value Theorem Note that in the exam, you may be ask to state the theorems you learned in this class.