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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 solution to the normal equations is considered the least-squares solution because it satisfies the necessary and sufficient condition for minimizing the residual norm $\|\vec{b} - A\vec{x}\|$. The derivation shows that minimizing this norm is equivalent to requiring the residual $A\vec{x}^* - \vec{b}$ to be orthogonal to the column space $C(A)$.

Conditions: The original system $A\vec{x}=\vec{b}$ may be inconsistent; $A^T A \vec{x}^* = A^T \vec{b}$ has a solution

The solution to the normal equations is considered the least-squares solution because it satisfies the necessary and sufficient condition for minimizing the residual norm $\|\vec{b} - A\vec{x}\|$. The derivation shows that minimizing this norm is equivalent to requiring the residual $A\vec{x}^* - \vec{b}$ to be orthogonal to the column space $C(A)$.

Conditions: The original system $A\vec{x}=\vec{b}$ may be inconsistent; $A^T A \vec{x}^* = A^T \vec{b}$ has a solution

The definition requires $x \neq a$ (expressed as $0 < |x - a|$) because the limit describes the behavior of the function *as it approaches* the point $a$, not its value *at* the point $a$. The function might be undefined at $x = a$, or its value $f(a)$ might differ from the limit $L$.

Conditions: The limit is concerned with the trend of $f(x)$ near $a$.; $f(a)$ may be undefined or discontinuous at $a$.

The definition requires $x \neq a$ (expressed as $0 < |x - a|$) because the limit describes the behavior of the function *as it approaches* the point $a$, not its value *at* the point $a$. The function might be undefined at $x = a$, or its value $f(a)$ might differ from the limit $L$.

Conditions: The limit is concerned with the trend of $f(x)$ near $a$.; $f(a)$ may be undefined or discontinuous at $a$.