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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 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 $f(a) \neq \lim_{x \to a} f(x)$.

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 $f(a) \neq \lim_{x \to a} f(x)$.