Calculus: Continuity, Discontinuity, and Limits

Practice flashcards covering key terms, conditions of continuity, types of discontinuity, greatest integer function, IVT, trig continuity, and infinite limits from Lessons 1.4 and 1.5.

Continuous Function (Informal)

A function whose graph can be drawn in a single continuous motion without lifting your pencil.

Continuity at a Point

A function $f(x)$ is continuous at $x = c$ if and only if three conditions are met: (1) $f(c)$ exists, (2) $\lim_{x \to c} f(x)$ exists, and (3) $\lim_{x \to c} f(x) = f(c)$.

Removable Discontinuity

A type of discontinuity represented by an open hole in the graph, which occurs at values where common factors in the numerator and denominator are canceled out.

Non-Removable Discontinuity

A type of discontinuity represented by a gap or a vertical asymptote, occurring at values where a factor in the denominator cannot be canceled out.

Limit of $\frac{|x|}{x}$ as $x \to 0$

An exception rule where the left-hand limit is $\lim_{x \to 0^-} \frac{|x|}{x} = -1$ and the right-hand limit is $\lim_{x \to 0^+} \frac{|x|}{x} = 1$, making the overall limit $\text{DNE}$.

Greatest Integer Function

A step function denoted by $[[x]]$ that evaluates to the largest integer less than or equal to $x$, always rounding left to the previous integer on a number line.

Intermediate Value Theorem (IVT)

A theorem stating that if a function is continuous on a closed interval $[a, b]$, it is guaranteed to touch all $y$-values between $f(a)$ and $f(b)$.

Continuity of Sine and Cosine

The trigonometric functions $\sin(x)$ and $\cos(x)$ are continuous everywhere on the real numbers ($\mathbb{R}$).

Discontinuity of Tangent and Secant

The trigonometric functions $\tan(x)$ and $\sec(x)$ are discontinuous at $x = \frac{\pi}{2} + k\pi$ where $k$ is an integer.

Discontinuity of Cosecant and Cotangent

The trigonometric functions $\csc(x)$ and $\cot(x)$ are discontinuous whenever $\sin(x) = 0$ (at $x = k\pi$ where $k$ is an integer).

Infinite Limit

A limit of the form $\lim_{x \to c} f(x) = +\infty$ or $\lim_{x \to c} f(x) = -\infty$, occurring when a function increases or decreases without bound as $x$ approaches $c$.

Vertical Asymptote

A vertical line $x = c$ where a function increases or decreases without bound, representing a non-removable discontinuity caused by a remaining denominator factor.