Why does a mismatch between left and right limits prevent continuity at a point?

At an interior point $a$, a function is continuous if it passes three tests: $f(a)$ is defined, the two-sided limit $\lim_{x\to a}f(x)$ exists, and this limit equals $f(a)$. The existence of the two-sided limit requires the left and right limits to agree.

Conditions: The point $a$ is an interior point of the function's domain.; The two-sided limit requires agreeing left and right limits.

Continuity can fail in three ways: a missing value violates the first test (the function is undefined); a jump violates the existence of a common two-sided limit (left and right limits differ); and a removable discontinuity occurs when the limit exists but differs from the assigned function value.

Conditions: The analysis applies to points where continuity is being tested.; A jump specifically means the left and right limits do not agree.; A removable discontinuity means the limit exists but does not equal $f(a)$.

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 $a$.