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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 $x$

To verify an antiderivative, you differentiate the proposed result. If the derivative of the proposed function equals the original integrand, then the proposal is correct.

Conditions: The proposed function is differentiable.; The original integrand is defined on the domain of interest.

To verify an antiderivative, you differentiate the proposed result. If the derivative of the proposed function equals the original integrand, then the proposal is correct.

Conditions: The proposed function is differentiable.; The original integrand is defined on the domain of interest.