---
tag: 063H5KJC69
kind: statement
role: theorem
name: First fundamental theorem of calculus
msc: [26A42]
uses: [0N91N98EG5, 05RCDKHZM6, 0F1WR9C83K]
sources:
  - url: https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus
    locator: First part
---

## Statement

Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be Riemann integrable.
Define $F(x) = \int_a^x f(t) \, dt$ for $x \in [a, b]$.
Then $F$ is continuous on $[a, b]$.
If $f$ is continuous at $c \in (a, b)$, then $F$ is differentiable at $c$ and $F'(c) = f(c)$.

## Remarks

If $|f| \le M$, then $|F(y) - F(x)| \le M |y - x|$. So $F$ is Lipschitz continuous.
By [[additivity|05RCDKHZM6]], $\frac{F(c + h) - F(c)}{h}$ is the average of $f$ over the interval between $c$ and $c + h$. Continuity at $c$ makes this average tend to $f(c)$.
Mathlib states a strict form as `intervalIntegral.integral_hasStrictDerivAt_of_tendsto_ae_right`.

## Explanations

### L1 human

Let the area under the graph from $a$ to $x$ be a function of $x$.
Where $f$ is continuous, this area grows at the rate $f(x)$.