Characteristic function of the uniform law

The nonnegative Borel density defines a measure, and its mass is $(b-a{)}^{-1}{\int }_a^{b}1dx=1$. The primitive $x$ and the integral bridge give this normalization. The density integration identity, applied to positive and negative parts of cosine and sine, yields $\phi (t)=(b-a{)}^{-1}{\int }_a^b(\cos (tx)+i\sin (tx))dx$; all four parts are integrable because the interval is finite and their absolute values are at most one.

For $t\mathrm{\ne }0$, the primitives are $\sin (tx)\mathrm{/}t$ for cosine and $-\cos (tx)\mathrm{/}t$ for sine, by the chain rule. Their derivatives are continuous on $[a,b]$, so FTC and the bridge give $\phi (t)=\frac{\sin (tb)-\sin (ta)-i(\cos (tb)-\cos (ta))}{t(b-a)}=\frac{e^{itb}-e^{ita}}{it(b-a)}\mathrm{.}$ At $t=0$ the integral of the constant one equals one. Continuity of characteristic functions then proves the claimed extension. The endpoints of $[a,b]$ have zero density measure, so using an open or half-open interval gives the same law. The hypothesis $a<b$ prevents division by zero; when $a=b$ this density is not defined, though the distinct Dirac law at $a$ has transform $e^{ita}$. The stated AC is spent on the compact integration bridge and its continuous-integrand prerequisites.