Singularity

In calculus, what is a singularity?

In calculus (and more broadly in mathematical analysis), a singularity of a function is a point at which the function ceases to be “well‑behaved” in some sense—typically because it is not defined there or its value (or derivative) “blows up” to infinity. More formally:

• Definition: A point $a$ is called a singularity of a real‑ or complex‑valued function $f(x)$ if $f$ is not analytic (i.e.\ not representable by a convergent power series) in any neighborhood of $a$.

  • In elementary (real) calculus this usually means $f$ is not defined at $a$, or $\lim_{x\to a}f(x)$ does not exist as a finite number.

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Types of Singularities (Real Calculus)

1. Removable singularity − The limit $\lim_{x\to a}f(x)$ exists (finite), but $f(a)$ is either undefined or defined to be something else. − Example:

   $$ f(x) = \frac{\sin x}{x} \quad\text{has}\quad \lim_{x\to0}\frac{\sin x}{x} = 1, $$

   but $f(0)$ is undefined. We “remove” the singularity by defining $f(0)\!=\!1$.

2. Infinite (or pole‑type) singularity − As $x\to a$, $\lvert f(x)\rvert\to\infty$. − Example:

   $$ f(x) = \frac{1}{x} \quad\text{blows up at}\quad x=0, $$

   so $x=0$ is an infinite singularity (often called a “pole” in complex analysis).

3. Jump (or essential discontinuity) − The one‑sided limits $\lim_{x\to a^+}f(x)$ and $\lim_{x\to a^-}f(x)$ both exist but are unequal (so the two‑sided limit doesn’t exist). − Example:

   $$ f(x)= \begin{cases} 1, & x<0,\\ 2, & x\ge0, \end{cases} $$

   has a jump at $x=0$.

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Singularities in Complex Analysis

When you extend to complex functions $f(z)$, singularities are classified more finely:

• Removable singularity: $\lim_{z\to a}f(z)$ exists; you can redefine $f(a)$ to make it analytic.
• Pole: $\lvert f(z)\rvert\to\infty$ as $z\to a$. A pole of order $n$ means $f(z)\sim (z-a)^{-n}$ near $a$.
• Essential singularity: Neither removable nor pole; $f(z)$ exhibits wild oscillation (Casorati–Weierstrass theorem).

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Why Singularities Matter

• Asymptotic behavior: Vertical asymptotes in graphs arise from infinite singularities.
• Integration: When performing improper integrals, singularities determine convergence or divergence.
• Series expansions: Analytic (power‑series) techniques break down at singularities; the distance to the nearest singularity governs a series’ radius of convergence.

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Key takeaway: A singularity is simply a point where a function “fails” to be nice—typically by being undefined or unbounded there. Understanding its type (removable, infinite/pole, jump or essential) tells you how you might “fix” it, how the graph behaves near it, and how analytic methods apply around it.