3 Step Continuity Test, Discontinuity, Piecewise Functions & Limits | Calculus

Continuity at a point requires the two-sided limit to exist and equal the function value. The two-sided limit exists only if the left-hand limit and the right-hand limit are equal.

Conditions: The point is an interior point of the domain.; The left-hand limit and right-hand limit are finite but unequal.

A jump discontinuity is identified when the left-hand limit and the right-hand limit at a point exist but are not equal. Because the one-sided limits differ, the two-sided limit does not exist, preventing the function from being continuous at that point.

Conditions: The function is defined on both sides of the point.; The left-hand limit and right-hand limit both exist as finite numbers.; The left-hand limit does not equal the right-hand limit.

A removable discontinuity occurs when the two-sided limit exists at a point, but the function is either undefined there or the function value does not equal the limit. Specifically, if the left and right limits are equal (e.g., both equal 3) but $f(a)$ is different (e.g., 5), the discontinuity is removable.

Conditions: The left-hand limit and right-hand limit at the point are equal.; The two-sided limit exists.; Either $f(a)$ is undefined or $f(a) \neq \lim_{x \to a} f(x)$.

For a function to be continuous at an interior point $a$, three conditions must be met: first, the function value $f(a)$ must be defined; second, the two-sided limit $\lim_{x \to a} f(x)$ must exist; and third, this limit must equal the function value $f(a)$. When checking the limit for piecewise functions, you must evaluate the correct branch on each side of $a$.

Conditions: The point $a$ is an interior point of the domain.; The function is defined piecewise or has branches around $a$.

A removable discontinuity can be restored to continuity by redefining the function value at the point of discontinuity to equal the existing two-sided limit. Since the limit already exists (left and right limits are equal), setting $f(a) = \lim_{x \to a} f(x)$ satisfies all three conditions for continuity.

Conditions: The two-sided limit exists at the point $a$.; The current function value $f(a)$ is either undefined or different from the limit.

A two-sided limit $\lim_{x \to a} f(x)$ is considered to exist if and only if both the left-hand limit $\lim_{x \to a^-} f(x)$ and the right-hand limit $\lim_{x \to a^+} f(x)$ exist and are equal to the same finite value. If they are unequal, the two-sided limit does not exist.

Conditions: The function is defined in a neighborhood around $a$ (excluding possibly $a$ itself).; We are evaluating the limit as $x$ approaches $a$.

The key difference lies in the existence of the two-sided limit. In a jump discontinuity, the left and right limits exist but are unequal, so the two-sided limit does not exist.

Conditions: Comparing two types of discontinuities at an interior point.; One-sided limits are finite in both cases.

Piecewise functions are defined by different formulas on different intervals. To determine the limit as $x$ approaches a boundary point $a$, you must use the formula valid for $x < a$ for the left-hand limit and the formula valid for $x > a$ for the right-hand limit.

Conditions: The function is defined piecewise.; The point $a$ is a boundary between branches.