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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.

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.

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 $f(a)$.

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 $a$.

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 $a$.