On pg. 57 of Principles of Quantum Mechanics, the author considers a vector space of "infinite dimensional vectors", conceived of as a vector space of functions defined on some closed interval. The inner product $⟨f,g⟩$ is defined as
$$ ⟨f|g⟩ = ∫_a^b f^*(x)g(x)dx $$
while a basis $|x_i⟩$ is defined as a function $x_i(x)$ s.t. $x_i(x) = 0$ everywhere except for $x_i(x_i) = 1$ (the author doesn't actually explicitly state this, but this I'm assuming is what he means?). At this point, it seems to me that
$$ ⟨x_i|x_i⟩ = ∫_a^b x_i^*(x)x_i(x)dx = 0 $$
since the integral of any function 0 almost everywhere is 0. But instead the author goes on to derive that the value is actually $δ(x_i-x_i) = ∞$:
This seems to me to be highly problematic since since an inner product is supposed to be a scalar function which returns scalar values, and $∞ ∉ \mathbb{C}$ is not scalar. In fact even the notion of $⟨x_i|f⟩$ seems to me to be confused, since
$$ ⟨x_i|f⟩ = ∫_a^b x_i^*(x)f(x)dx = 0 $$
since again the integral of a function 0 everywhere except for $x^*_i(x_i)f(x_i) = 1*f(x_i) = f(x_i)$ is still just $0$. Yet the author wants to say that this integral evalutes to $f(x_i)$?
Further to @CosmasZachos's answer, nothing in the excerpt says $x_i(x_i)=1$, i.e. that $x_i(x)=\delta_{x,\,x_i}$ is a Kronecker delta. This is a misconception on your part. It really should be a Dirac delta, so$$\langle x_i|x_j\rangle=\int_a^b\delta(x-x_i)\delta(x-x_j)dx=\delta(x_i-x_j)$$ provided $x_i\in(a,\,b)$ (otherwise the RHS is $0$ instead).