Unit 1 Master Guide: Continuity in Calculus

Continuity at a Point

A function is continuous at a point if three conditions are met: the function value is defined, the limit exists, and the limit equals the function value.

Definition of Continuity Checklist

A list of the three conditions required for a function to be continuous at a specific point.

Condition 1 for Continuity

f(c) must be defined: The function must have a value at x=c.

Condition 2 for Continuity

lim_{x -> c} f(x) must exist: The left and right limits must be equal.

Condition 3 for Continuity

The limit must equal the function value: lim_{x -> c} f(x) = f(c).

Discontinuous

A function is discontinuous if any of the conditions for continuity fails.

Removable Discontinuity

Occurs when the limit exists but does not equal the function value; visually represented by a hole.

Jump Discontinuity

A type of discontinuity where the left-hand limit and right-hand limit exist but are not equal; visually represented by a jump.

Infinite Discontinuity

Occurs when the function approaches positive or negative infinity causing a vertical asymptote.

Oscillating Discontinuity

A rare type of discontinuity where the function oscillates infinitely fast as it approaches a point.

Continuous on Open Intervals

A function is continuous on an open interval (a, b) if it is continuous at every point in that interval.

Continuous on Closed Intervals

A function is continuous on a closed interval [a, b] if it's continuous on (a, b) and continuous from the right at a and from the left at b.

One-Sided Continuity

The concept of continuity that must be checked at the endpoints of a closed interval.

Determining Removable Discontinuities

Typically involves checking for a removable discontinuity if the indeterminate form 0/0 occurs when evaluating the limit.

Algebraic Approach to Discontinuities

Involves simplifying expressions to identify and remove removable discontinuities.

Example of Removable Discontinuity

For f(x) = (x^2 - 9)/(x - 3), there is a hole at x=3 that can be removed by redefining the function at that point.

Common Mistake 1

Confusing the existence of limits with functions being continuous; all three conditions must be checked.

Common Mistake 2

Assuming that a zero in the denominator automatically indicates a vertical asymptote; must check if it causes a removable discontinuity.

Common Mistake 3

Ignoring the need for one-sided limits at the endpoints of a closed interval.

Vertical Asymptote

Occurs when the function approaches infinity as x approaches a certain value.

Undefined Function Value

Condition 1 fails if a function is not defined at x=c.

Piecewise Functions Containing Discontinuities

Often exhibit jump discontinuities between segments.

Behavior of Continuous Functions

Polynomials and trigonometric functions are continuous everywhere in their domains.

Limit Notation for Continuity

For continuity, lim_{x->c} f(x) must equal f(c) as part of the three defining conditions.

Analyzing Function Behavior

Understanding continuity and discontinuity types helps analyze overall behavior of functions in calculus.