What would you like to understand?

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

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 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 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.