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

$f(x) = \frac{p(x)}{q(x)}$, where p(x) and q(x) are polynomials.

$f(a) = \lim_{x \to a} f(x)$, where 'a' is the x-value of the discontinuity.

If $f(x) = \frac{(x-c)g(x)}{(x-c)}$, then the simplified function is $g(x)$ for $x \neq c$.

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

$f(x) = \begin{cases} f_1(x), & x \in D_1 \\ f_2(x), & x \in D_2 \\ ... \end{cases}$

Solve $q(x) = 0$ where $f(x) = \frac{p(x)}{q(x)}$

$f(x) = \begin{cases} original, & x \neq a \\ \lim_{x \to a} original, & x = a \end{cases}$

$f(x) = \frac{(x-a)g(x)}{(x-a)} = g(x)$

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

A point where the limit exists, but the function is undefined or has a different value.

A function is continuous at x=a if the limit as x approaches a exists, f(a) is defined, and the limit equals f(a).

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

The value that a function approaches as the input approaches some value.

The function does not have a value at that specific x-value.

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

A function that can be drawn without lifting your pencil.

The value the function approaches as x approaches a value from the left.

The value the function approaches as x approaches a value from the right.

Redefining the function's value at the point of discontinuity to be equal to the limit at that point, thus making the function continuous.

It involves redefining the function at the point of discontinuity to equal the limit at that point, effectively 'filling the hole' and making the function continuous.

Factoring allows you to simplify rational functions and cancel out common factors that cause discontinuities, revealing the simplified function and the location of the hole.

The left-hand limit and the right-hand limit at the boundary must exist and be equal to each other, and the function's value at the boundary must also be equal to this limit.

Because the limit exists at that point, and by redefining the function at that point, we can make the function continuous.

The graph appears continuous except for a single point (a 'hole') where the function is undefined.

Because the function's behavior may be different on either side of the boundary point, and continuity requires these limits to match.

Factor the numerator and denominator, cancel any common factors, and identify the x-values that were canceled out.

The limit tells us where the function 'should' be, even if it isn't defined there. This allows us to redefine the function and remove the discontinuity.

Graphs provide a visual representation of where a function might have breaks, jumps, or holes, making it easier to spot potential discontinuities.

Setting the denominator equal to zero helps to find potential points of discontinuity, and setting the numerator equal to zero helps to find the x-intercepts of the function.