Are topological defects such as domain walls really discontinuous?

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Are topological defects really discontinuous
Are topological defects such as domain walls really discontinuous (implying infinite slopes of the fields) or only approximately ?
 
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I don't know topology, but I do have a question. How can "discontinuous" not be a binary choice? What does "approximately discontinuous" mean?
 
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That's basically the definition of a topological defect, so yes, in a way.

They're not necessarily boundaries, though. Cosmic strings, for example, are one-dimensional structures where within the string the symmetry breaking that resulted in our observable universe is stuck in a different local minimum of symmetry of higher rest energy.

They can also be two-dimensional boundaries between different regions of distinct symmetry breaking (domain walls). If the rest energy differs, these domain walls will move at close to the speed of light.

As far as the width/size of the discontinuity, a quick Google search suggests that it depends upon the specific theory, but typical predictions around $10^{-32}$ meters (a thousand times the Planck length). I believe this is a thickness set by the energy at which the phase transition happens.