Continuity - Calculus I - Lecture Slides, Slides of Calculus

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1.5 Continuity

Most of the techniques of calculus require that functions be continuous. A function is continuous if you can draw it in one motion without picking up your pencil.

Definition: A function f is continuous at a number a if

(the limit is the same as the value of the function).

This function is discontinuous (has discontinuities) at x=1 and x=2.

It is continuous everywhere else (^1 2 3 4) on the interval [0,4].

1

2

lim f ( x ) f ( a^ )

x a

= →

• Definition: A function f is continuous from the right at a number a if

and f is continuous from the left at a number a if

• Definition: A function f is continuous on an interval if it is continuous at every number in the interval.

Examples on the board.

lim f ( x ) f ( a^ ) x a

→^ +

lim f ( x ) f ( a^ ) x a

→ −

• Theorem: If f and g are continuous at a and c is a constant, then the following functions are also continuous at a :
• Theorem: The following types of functions are continuous at every number in their domains: polynomials, rational functions, root functions, trigonometric functions.

Example: The function

is continuous on the intervals [0,2) and (2,∞)

4. 5. if ( ) 0

≠

• −

g a g

f fg

f g f g cf

x x

x (^) 2 2

(^3) + −