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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.

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 ensures that as the interval width $h$ approaches zero, the average value of $f$ over $[x, x+h]$ converges to the instantaneous value $f(x)$. Without continuity, the local behavior might oscillate wildly or have jumps, preventing the limit of the difference quotient from settling on a single well-defined value $f(x)$.

Conditions: $A(x) = \int_a^x f(t) dt$; $f$ is continuous at $x$

Continuity ensures that as the interval width $h$ approaches zero, the average value of $f$ over $[x, x+h]$ converges to the instantaneous value $f(x)$. Without continuity, the local behavior might oscillate wildly or have jumps, preventing the limit of the difference quotient from settling on a single well-defined value $f(x)$.

Conditions: $A(x) = \int_a^x f(t) dt$; $f$ is continuous at $x$

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 $f(a)$.

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.

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 $a$.

The simplified functions agree with the original expressions near the target point because the algebraic manipulations (factoring, conjugate multiplication, clearing fractions) are equivalences valid for all $x$ in the domain except possibly the target point itself. Limits describe the behavior of a function as the input approaches a value, not the value of the function at that point.

Conditions: The target point is excluded from the domain of the original expression.; The algebraic simplification is valid for all $x$ near the target point ($x \ne$ target).; The limit exists for the simplified function.

The simplified functions agree with the original expressions near the target point because the algebraic manipulations (factoring, conjugate multiplication, clearing fractions) are equivalences valid for all $x$ in the domain except possibly the target point itself. Limits describe the behavior of a function as the input approaches a value, not the value of the function at that point.

Conditions: The target point is excluded from the domain of the original expression.; The algebraic simplification is valid for all $x$ near the target point ($x \ne$ target).; The limit exists for the simplified function.