Some define the Dirac Delta Function as:
$$\int_{-\infty}^{\infty}\delta(x)f(x)\ dx=f(0)$$
For every continuous function $f$. In some books, I've noticed a different definition of $\delta(x)$ as an operation that satisfies the following two conditions:
$$\int_{-\infty}^{\infty}\delta(x)\ dx=1\quad\text{and}\quad\forall x\neq0:\delta(x)=0$$
Are the two definitions the same?
First of all, $\delta $ is not a function ! There are no function s.t. $$\int_{\mathbb R}\delta (x)=1\quad \text{and}\quad \forall x\neq 0, \delta (x)=0.$$
The second definition is rather not correct strictly speaking, but should be understood as $$\int_{\mathbb R}\delta (x)\,\mathrm d x=1\quad \text{and}\quad \int_A \delta (x)\,\mathrm d x=\begin{cases}1&0\in A\\ 0&\text{otherwise}\end{cases}.$$
So, no they are not the same. But they are indeed equivalents.