Basic properties of characteristic functions

Write $\phi (t)=\int e^{itx}\mu (dx)$ with $\mu (\mathbb{R})=1$. At zero the integrand is one, so $\phi (0)=1$. The triangle inequality gives $\mathrm{\mid }\phi (t)\mathrm{\mid }\le \int \mathrm{\mid }{e}^{itx}\mathrm{\mid }d\mu =1$.

Writing the integral componentwise, $\phi (-t)=\int \cos (tx)d\mu -i\int \sin (tx)d\mu =\overline{\phi (t)}$. Linearity and exponential addition further give $\mathrm{\mid }\phi (t+h)-\phi (t)\mathrm{\mid }\le \int \mathrm{\mid }{e}^{itx}(e^{ihx}-1)\mathrm{\mid }d\mu =\int \mathrm{\mid }{e}^{ihx}-1\mathrm{\mid }d\mu \mathrm{.}$

For real $u$, apply the mean value theorem separately to sine and cosine between $0$ and $u$. Their derivatives have absolute value at most one by Euler's formula, so $\mathrm{\mid }\sin u\mathrm{\mid }\le \mathrm{\mid }u\mathrm{\mid }$ and $\mathrm{\mid }\cos u-1\mathrm{\mid }\le \mathrm{\mid }u\mathrm{\mid }$. Thus $\mathrm{\mid }{e}^{iu}-1\mathrm{\mid }\le \min (2,2\mathrm{\mid }u\mathrm{\mid })$. For each positive integer $n$, define $g_n(x)=\min (2,2\mathrm{\mid }x\mathrm{\mid }\mathrm{/}n)$. This measurable sequence tends pointwise to zero and is dominated by the integrable constant $2$, so DCT gives $\int g_{n}d\mu \to 0$. Whenever $\mathrm{\mid }h\mathrm{\mid }\le 1\mathrm{/}n$, step 1.2 bounds $\mathrm{\mid }\phi (t+h)-\phi (t)\mathrm{\mid }$ by $\int g_{n}d\mu$, for every $t$. Given $\epsilon >0$, take the least positive integer $n$ with this integral less than $\epsilon$. Then $\delta =1\mathrm{/}n$ proves uniform continuity. This uses a prescribed sequence and no choice of a sequence of counterexamples.