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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 $f(a)$.

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 $f(a)$.

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.

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.