Consider the function $\phi$ defined by $$\phi(t) = \left\{\begin{array}{ccc}t^2+1-2|t|& \rm for &|t|\le 1- \frac{1}{\sqrt{2}}\\ \frac{1}{2}&\rm for& |t|> 1- \frac{1}{\sqrt{2}}\end{array}\right. $$ Is $\phi$ a characteristic function?
I assume it is not, as it do not have a derivative at $t=1- \frac{1}{\sqrt{2}}$, but how to prove that?
If we define
$$\varphi_1(t) := \begin{cases} 2t^2-4 |t| +1, & |t| \leq 1- \frac{1}{\sqrt{2}} \\ 0, & |t| > 1- \frac{1}{\sqrt{2}}\end{cases},$$
then $\varphi_1$ is an even continuous real-valued function which satisfies $$\varphi_1(0)=1 \quad \text{and} \quad \lim_{|t| \to \infty} \varphi_1(t)=0.$$ Moreover, $\varphi_1|_{(0,\infty)}$ is convex. It follows from Polya's theorem that $\varphi_1$ is a characteristic function. On the other hand,
$$\varphi_2(t) := 1$$
is clearly a characteristic function. This implies that
$$\varphi(t) = \frac{1}{2} \varphi_1(t) + \frac{1}{2} \varphi_2(t)$$
is a characteristic function.