AP Calculus AB Unit 1 Notes: Understanding Continuity

Continuity (intuitive idea)

A function is continuous at a point if its graph has no break there—informally, you can draw it without lifting your pencil (but the formal definition uses limits and function values).

Discontinuity

A point where a function fails to be continuous; determined by limit behavior and the function value, not just how the graph looks.

Removable discontinuity

A discontinuity where the two-sided limit exists and is finite, but the function value is missing (undefined) or not equal to the limit (often appears as a “hole” or misplaced dot).

Jump discontinuity

A discontinuity where both one-sided limits exist as finite numbers but are not equal, so the two-sided limit does not exist.

Infinite discontinuity

A discontinuity where the function grows without bound near a point (limit goes to ±∞), typically producing a vertical asymptote.

Left-hand limit

(\lim_{x\to a^-} f(x)): the value(s) f(x) approaches as x approaches a from values less than a.

Right-hand limit

(\lim_{x\to a^+} f(x)): the value(s) f(x) approaches as x approaches a from values greater than a.

Two-sided limit

(\lim_{x\to a} f(x)): exists only when both one-sided limits exist and are equal.

Vertical asymptote

A vertical line (x=a) where function values blow up (approach ±∞), indicating an infinite discontinuity.

Formal definition of continuity at (x=a)

f is continuous at 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)).

“Perfect handshake” (limit-value match)

A way to remember continuity: the value the function approaches near a (the limit) must equal the value the function takes at a.

Misplaced dot (continuity failure)

When (\lim_{x\to a} f(x)) exists but (f(a)) is defined at a different value, causing a removable discontinuity.

Indeterminate form (0/0)

A substitution result that signals more algebraic work is needed (e.g., factoring or rationalizing); it does not automatically mean the limit does not exist.

Factor-and-cancel technique

A method to evaluate limits and detect removable discontinuities by factoring expressions and canceling common factors (valid for simplifying near the point, not at the point where the factor is zero).

Rationalizing (using a conjugate)

A technique for limits with radicals: multiply by the conjugate to eliminate radicals and simplify, often revealing a removable discontinuity.

Conjugate

For expressions like (\sqrt{u}-v), the conjugate is (\sqrt{u}+v); multiplying by it can simplify a radical expression.

Right-continuous at (x=a)

A one-sided continuity condition: (\lim_{x\to a^+} f(x)=f(a)), used at left endpoints or domain start points.

Left-continuous at (x=a)

A one-sided continuity condition: (\lim_{x\to a^-} f(x)=f(a)), used at right endpoints or domain end points.

Continuous on an open interval ((a,b))

A function is continuous on ((a,b)) if it is continuous at every point strictly between a and b.

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

Requires continuity on ((a,b)) plus right-continuity at a and left-continuity at b.

“Danger points” for continuity

Inputs where continuity may fail, such as zeros of denominators, invalid radical inputs (negative radicand), nonpositive log inputs, or piecewise switch points.

Rational function continuity rule

A rational function is continuous everywhere its denominator is not zero (discontinuities occur where the denominator equals 0).

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).

IVT sign-change (root guarantee)

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

Removing a discontinuity (redefining a point)

If (\lim{x\to a} f(x)) exists and is finite but f is not continuous at a, define a new value (g(a)=\lim{x\to a} f(x)) (changing only that point) to make the function continuous there.