A function is continuous on an interval if it is continuous at every point within that interval.

A function defined by multiple sub-functions, each applying to a certain interval of the main function's domain.

A limitation on the possible input values (x-values) for a function, often due to square roots or division by zero.

The value a function approaches as the input approaches a given value from the left side.

The value a function approaches as the input approaches a given value from the right side.

The left-hand limit, right-hand limit, and the function's value at that point must all be equal.

Functions that can be expressed as the quotient of two polynomials.

Functions containing a radical, such as a square root or cube root.

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

A function that can be written in the form (f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0), where n is a non-negative integer and the coefficients are constants.

\(\lim_{x\to a^{-}} f(x) = \lim_{x\to a^{+}} f(x) = f(a))

Check if (\lim_{x\to c^{-}} f(x) = \lim_{x\to c^{+}} f(x) = f(c))

\(\sqrt{z}) requires (z \geq 0)

\(\frac{1}{z}) requires (z \neq 0)

\(f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0)

Solve the inequality (g(x) \geq 0).

Solve the equation (g(x) = 0) and exclude those values from the domain.

If (f) is continuous on ([a, b]) and (k) is a number between (f(a)) and (f(b)), then there exists at least one (c) in ([a, b]) such that (f(c) = k).

If (\lim_{x \to a} f(x)) exists but is not equal to (f(a)), or (f(a)) is undefined, then there is a removable discontinuity at (x = a).

If (f(x)) is differentiable at (x=a), then (f(x)) is continuous at (x=a).

Polynomials are continuous everywhere because they have no breaks, jumps, or vertical asymptotes in their graphs.

Rational functions are continuous everywhere except where the denominator is zero, which creates a vertical asymptote or a hole.

You must verify that the left-hand limit, right-hand limit, and the function's value at that point are all equal to ensure there are no jumps or breaks.

Domain restrictions indicate points where the function is not defined, leading to discontinuities. These points must be excluded when determining continuity over an interval.

For a function to be continuous at a point, the limit as x approaches that point must exist and be equal to the function's value at that point.

A removable discontinuity occurs when a function has a hole at a point, meaning the limit exists, but the function is either undefined or has a different value at that point. It can be 'removed' by redefining the function at that point.

A non-removable discontinuity occurs when the limit does not exist at a point, such as at a vertical asymptote or a jump discontinuity. It cannot be 'removed' by redefining the function at that point.

Checking both limits is essential for determining continuity, especially in piecewise functions, because the function's behavior may differ on either side of a point.

Trigonometric functions like sine and cosine are continuous everywhere. Tangent, cotangent, secant, and cosecant are continuous on their domains, excluding points where they have vertical asymptotes.

Exponential functions are continuous everywhere. Logarithmic functions are continuous on their domains, which are typically (x > 0).