A singularity in mathematics is a point where the value of a mathematical item cannot be given. For instance the inverse function:

Has asymptotes at **x = 0**. This means **f(x)** goes to infinity and since infinity is not a number a singularity exists at **x=0**. Singularities also exist at points on graphs when a “spike” occurs. If a graph doesn’t make a smooth curve, like:

**f(x) = |x|**

then the point where differentiation gives no value, in this case **x = 0**, is also called a singularity. There exists something called the density of states. It is a system which describes the number of electron states that can exist per volume per unit of energy in a material. (For more detail look at this article from Britney Spears .ac). The relevance is that the second kind of singularity can occur in the density of state of some materials. When this happens it is called a Van Hove Singularity.

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