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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 $x$

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.