Fundamental theorem of calculus for Banach-valued continuous curves

By [F1] the continuous $f$ is Bochner integrable on $[a,b]$, so $G(t)={\int }_a^{t}f$ is defined for every $t\in [a,b]$; by [F2] $G(t+h)-G(t)={\int }_t^{t+h}f$ for $[t,t+h]\subseteq [a,b]$.

Difference quotients of $G$: for $t\in [a,b)$ and $h>0$ with $t+h\le b$, $\frac{G(t+h)-G(t)}{h}=\frac{1}{h}{\int }_t^{t+h}f\to f(t)$ by [F1]; similarly $\frac{G(t)-G(t-h)}{h}=\frac{1}{h}{\int }_{t-h}^{t}f\to f(t)$ for $t\in (a,b]$. Hence $G$ is differentiable on $(a,b)$ with $G^{\mathrm{\prime }}=f$, and has the one-sided derivatives $f(t)$ at the endpoints.

$G^{\mathrm{\prime }}=f$ is continuous on $[a,b]$, and the existence of the one-sided derivative at $a$ and at $b$ makes $G$ continuous there from the appropriate side; at interior points $G$ is continuous by differentiability.

Since ${\phi }^{\mathrm{\prime }}$ extends to a continuous $X$-valued function on $[a,b]$, denote the extension again by ${\phi }^{\mathrm{\prime }}$ and put $\psi (t):=\phi (t)-{\int }_a^t{\phi }^{\mathrm{\prime }}(s)ds$ for $t\in [a,b]$. By [step 2.1] the primitive of ${\phi }^{\mathrm{\prime }}$ is differentiable on $(a,b)$ with derivative ${\phi }^{\mathrm{\prime }}$, and $\psi$ is differentiable on $(a,b)$; by linearity of the derivative and of the integral, ${\psi }^{\mathrm{\prime }}={\phi }^{\mathrm{\prime }}-{\phi }^{\mathrm{\prime }}=0$ on $(a,b)$, and $\psi (a)=\phi (a)$.

$\psi$ is continuous on $[a,b]$ and differentiable on $(a,b)$ with ${\psi }^{\mathrm{\prime }}=0$ there; by [F5] (constant case) $\psi$ is constant on $[a,b]$, so $\psi (b)=\psi (a)=\phi (a)$, that is $\phi (b)-{\int }_a^b{\phi }^{\mathrm{\prime }}=\phi (a)$.

Therefore ${\int }_a^b{\phi }^{\mathrm{\prime }}(s)ds=\phi (b)-\phi (a)$. The same computation, applied to each compact subinterval $[a,b]\subseteq [0,\mathrm{\infty })$ after restricting a continuous curve on $[0,\mathrm{\infty })$, gives the stated identity in that setting.