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Ionic bonds have covalent properties that depend on the polarizability of the anion and the cation's polarizing power. Ionic bonds have more covalent character when the anion is more polarizable or when the cation has high polarizing power.
I think the idea is that bonds are not binary. For example, like gender, bonds lie on a spectrum of ionic and covalent character. Therefore, they are not always equally sharing (covalent) or completely giving up electrons (ionic), so they have some characteristics of both.
When ionic bonds form, an electron cloud forms between the two atoms. When opposite charged atoms approach each other, the attraction and repulsion of opposite and alike charges leads to polarization in the anion. A large amount of polarization increases the covalent character, while less polarization has a smaller amount of covalent character.
The cations in ionic compounds have a certain polarizing power: their positive charge makes them pull on the electrons of neighboring anions (that we describe as being polarizable to a certain extent). Because of this, there's a distortion in the electron cloud of the anion, making it so that there is some covalent character (electrons being pulled between two ions rather than electrons being just strictly transferred).
An ionic compound has at least two atoms. Even in the atom that donated an electron, the nucleus still exerts a small attractive force on the electrons of the other atom(s). This causes them to be pulled slightly in one direction, which means they are slightly covalent.
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