1. Math 1304 Calculus I 2.5 – Continuity

2. Definition of Continuity Definition: A function f is said to be continuous at a point a if and only if the limit of f as x approaches a is the value f(a) of f at a. That is:

3. Definition of Left Continuity Definition: A function f is said to be left continuous at a point a if and only if the limit of f as x approaches a from the left is the value f(a) of f at a.

4. Definition of Right Continuity Definition: A function f is said to be right continuous at a point a if and only if the limit of f as x approaches a from the right is the value f(a) of f at a.

5. Definition of continuity on an interval Definition: A function f is said to be continuous on an interval if and only if it is continuous at every point on the interval.

6. Theorems on Continuity Theorem: If f and g are functions that are continuous at a point a, then the sum, difference, multiple by constant are continuous at that point. The quotient is continuous if the denominator is not zero at the point. Proof: Use limit formulas – rules introduced in section 2.3

7. Theorems on Continuity Theorem: Polynomials are continuous at every point Proof: Write any polynomial as a sum of products. Use the sum rule and the product rule for limits.

8. Theorems on Continuity Theorem: Rational functions are continuous where defined. Proof: Use the quotient rule for limits.

9. Theorems on Continuity Theorem: Roots, trigonometry, exponential, logarithmic, and inverse trigonometric functions are continuous where defined. Proof: Use limit formulas for each type. These require separate proofs that are beyond this course.

10. Theorems on Continuity Theorem: If f is a continuous function at a point a, and g is is a function that is continuous at the point b=f(a), then the composite g○f is continuous at a. Proof: Prove a limit formula (Appendix F)

11. Intermediate value theorem. Theorem: Suppose that f is continuous on the closed interval [a, b] and let N be any number between f(a) and f(b). Then there exists a number c in (a,b) such that f(c)=N. Example: page 134, #47