AP Calculus BC Unit 1 Notes: Understanding Continuity and Its Consequences

Continuity (informal)

A function is continuous if its graph can be drawn without lifting your pencil (no breaks, holes, or jumps).

Discontinuity

A point where a function fails to be continuous; the “unbroken drawing” idea breaks down and is diagnosed using limits (often one-sided limits).

One-sided limit

A limit taken from only one side of a point, written as (\lim{x\to a^-} f(x)) (left) or (\lim{x\to a^+} f(x)) (right).

Removable discontinuity

A discontinuity at (x=a) where (\lim{x\to a} f(x)) exists (finite) but (f(a)) is undefined or (f(a)\neq \lim{x\to a} f(x)); graphically looks like a hole and can be fixed by redefining one point.

Hole (open circle)

The graphical appearance of a removable discontinuity: the curve approaches a single y-value at (x=a) but the function value at that point is missing or different.

Jump discontinuity

A discontinuity where both one-sided limits exist and are finite, but they are not equal, so the two-sided limit does not exist (the graph “jumps”).

Left-hand limit

The value (f(x)) approaches as (x) approaches (a) from the left, written (\lim_{x\to a^-} f(x)).

Right-hand limit

The value (f(x)) approaches as (x) approaches (a) from the right, written (\lim_{x\to a^+} f(x)).

Two-sided limit

The limit (\lim_{x\to a} f(x)), which exists only if the left-hand and right-hand limits both exist and are equal (and finite in AP Calculus usage).

Infinite discontinuity

A discontinuity where (f(x)) grows without bound (toward (\infty) or (-\infty)) as (x\to a), often associated with a vertical asymptote.

Vertical asymptote

A vertical line (x=a) where the function becomes unbounded, e.g., (\lim_{x\to a} f(x)=\pm\infty); indicates an infinite discontinuity.

Oscillating discontinuity

A limit failure where the function oscillates infinitely near a point and does not approach a single value (e.g., (\sin(1/x)) near (x=0)).

Continuity at a point

A function is continuous at (x=a) if (1) (f(a)) is defined, (2) (\lim{x\to a} f(x)) exists, and (3) (\lim{x\to a} f(x)=f(a)).

Right-continuous

At (x=a), a function is right-continuous if (\lim_{x\to a^+} f(x)=f(a)) (used at left endpoints of intervals).

Left-continuous

At (x=b), a function is left-continuous if (\lim_{x\to b^-} f(x)=f(b)) (used at right endpoints of intervals).

Continuous on a closed interval ([a,b])

Means continuous at every interior point (a<c<b), right-continuous at (a), and left-continuous at (b).

Domain restriction (continuity)

Continuity statements apply only where the function is defined; e.g., rational functions are continuous everywhere except where their denominator is zero.

Polynomial continuity

Polynomials are continuous for all real numbers.

Rational function continuity

A rational function is continuous at every (x) in its domain (i.e., at all real (x) where the denominator is not zero).

Composition rule for continuity

Sums, differences, products, and compositions of continuous functions are continuous wherever the resulting expression is defined.

Removing a discontinuity

Making a function continuous by modifying it—typically redefining (f(a)) to match a finite limit at (x=a), which only works for removable discontinuities.

Continuity fix condition

To make (f) continuous at (x=a), you must set (f(a)=\lim_{x\to a} f(x)) (the limit must exist and be finite).

Factor-cancel pattern (removable discontinuity)

A removable discontinuity often occurs when ((x-a)) cancels from numerator and denominator: (\frac{(x-a)g(x)}{(x-a)h(x)}), leaving a hole at (x=a) in the original function.

Intermediate Value Theorem (IVT)

If (f) is continuous on ([a,b]) and (N) is between (f(a)) and (f(b)), then there exists (c\in[a,b]) such that (f(c)=N).

Sign change implies a root (via IVT)

If (f) is continuous on ([a,b]) and (f(a)\cdot f(b)<0), then there exists (c\in[a,b]) with (f(c)=0) (IVT guarantees a zero).