Look for jumps, holes (open circles), or vertical asymptotes. These indicate points where the function is not continuous.

A smooth, unbroken curve indicates that the function is continuous over that interval. There are no sudden jumps or breaks.

A vertical asymptote indicates an infinite discontinuity. The function is not continuous at the x-value where the asymptote occurs.

An open circle indicates a removable discontinuity (a hole). The function is not defined at that specific x-value, or the limit does not equal the function value.

A jump in the graph indicates a jump discontinuity. The left-hand limit and the right-hand limit exist but are not equal.

Check if the pieces of the graph connect smoothly at the points where the function definition changes. There should be no gaps or jumps.

If a function is differentiable, its graph is smooth and continuous. There are no sharp corners, cusps, or discontinuities.

The graph of $f(x) = \frac{1}{x}$ has a vertical asymptote at x = 0, indicating a discontinuity at that point. The function is continuous everywhere else.

The graph of $f(x) = |x|$ is continuous everywhere. Although it has a sharp corner at x = 0, it is still continuous at that point.

Trace the graph from both the left and right sides towards the x-value of interest. If the y-values approach the same number, that number is the limit.

Continuity: Function has no breaks, jumps, or holes. | Differentiability: Function has a derivative at every point (smooth, no sharp corners or vertical tangents).

Removable: Limit exists, but doesn't equal f(c) or f(c) is undefined. | Jump: Left and right limits exist, but are not equal.

Continuous: Satisfies all three conditions for continuity at every point in its domain. | Discontinuous: Fails at least one of the three conditions for continuity at one or more points in its domain.

Left-hand limit: The value f(x) approaches as x approaches c from the left (x < c). | Right-hand limit: The value f(x) approaches as x approaches c from the right (x > c).

Continuity at a point: Function satisfies the three conditions for continuity at a specific x-value. | Continuity on an interval: Function is continuous at every point within that interval.

Rational function: Can have discontinuities where the denominator is zero. | Polynomial function: Continuous everywhere.

Vertical asymptote: Represents an infinite discontinuity where the function approaches infinity. | Hole: Represents a removable discontinuity where the function is undefined but the limit exists.

Direct substitution: Plug in the value directly into the function. Works for continuous functions. | Limit laws: Apply algebraic rules to simplify and evaluate limits, especially when direct substitution fails.

Intermediate Value Theorem: Guarantees a value between f(a) and f(b) for a continuous function on [a, b]. | Extreme Value Theorem: Guarantees a maximum and minimum value for a continuous function on [a, b].

Definition: Rigorous approach using limits and function values. | Graph: Visual approach identifying breaks, jumps, or holes.

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

A function is discontinuous at x=c if at least one of the three conditions for continuity is not met: f(c) is undefined, $\lim_{x \to c} f(x)$ does not exist, or $\lim_{x \to c} f(x) \neq f(c)$.

The left-hand limit is the value that f(x) approaches as x approaches c from values less than c, denoted as $\lim_{x \to c^-} f(x)$.

The right-hand limit is the value that f(x) approaches as x approaches c from values greater than c, denoted as $\lim_{x \to c^+} f(x)$.

A removable discontinuity occurs when $\lim_{x \to c} f(x)$ exists, but either f(c) is undefined or $\lim_{x \to c} f(x) \neq f(c)$. It can be 'removed' by redefining f(c).

A jump discontinuity occurs when the left-hand limit and the right-hand limit both exist but are not equal: $\lim_{x \to c^-} f(x) \neq \lim_{x \to c^+} f(x)$.

An infinite discontinuity occurs when the function approaches infinity (or negative infinity) as x approaches c from either the left or right, often associated with vertical asymptotes.

An oscillating discontinuity occurs when the function oscillates infinitely many times near a point, preventing the limit from existing.

The domain of a function is the set of all possible input values (x-values) for which the function is defined.

The range of a function is the set of all possible output values (y-values) that the function can produce.