Continuity and discontinuity

You’ve gotta dance like there’s nobody watching, love like you’ll never be hurt,sing like there’s nobody listening, and live like it’s heaven on earth, William W. Purkey.

A function f(x) is continuous at a point x = a if and only if the following three conditions are satisfied:

• i. It must be defined at that point, i.e., f(a) is defined.
• ii. Its limit must exist at the point, $\lim_{x \to a} f(x)$ exists. The left and right-hand limits are defined and equal.
• iii. The value of the function at that point must equal the value of the limit at that point, $\lim_{x \to a} f(x) = f(a).$

Geometrically, you can think of a function that is continuous as a function whose graph has no break in it. Intuitively, f is continuous at a if f(x) gets closer and closer to f(a) as x gets closer and closer to a. Formally, let f(x) be a function defined on an interval that contains x = a, then we say that f(x) is continuos at x = a, if

$\forall \epsilon>0, \exists \delta>0: |f(x)-f(a)|<\epsilon, whenever~ |x-a| < \delta.$

What is a Discontinuous Function?

A function is discontinuous if you cannot draw its graph without lifting your pen. The function’s graph does not form a continuos line, but instead has gaps such as jumps, holes, or breaks. More formally, a function is discontinuous at a point a if it fails to be continuous at a, that is, when it breaks any of the continuity criteria, such as, f(a) is not defined, $\lim_{x \to a} f(x)$ does not exist, or $\lim_{x \to a} f(x) ≠ f(a).$

Types of Discontinuity

There are different types of discontinuities:

• A jump discontinuity is a discontinuity where the value of a function ‘jumps’ from one value to another, that is, where the right hand limit and left hand limit both exist, but these limits are not equal to each other. Examples:

$f(x) = \begin{cases} x + 1, &x > 0\\\\ -2x + 2, &x < 0 \end{cases}$ -1.g.-

$\lim_{x \to 0^{+}} f(x) = 1 ≠ 2 = \lim_{x \to 0^{-}} f(x).$

$f(x) = \begin{cases} -x^{2} + 4, &x≤3\\\\ 4x - 8, &x > 3 \end{cases}$

$\lim_{x \to 3^{+}} f(x) = 4 ≠ -5 = \lim_{x \to 3^{-}} f(x).$

• A removable discontinuity is a discontinuity that results when the limit of a function exists but is not equal to the value of the function at the given point. It seems that the function is begging in agony to be redefined. f(x)=^sin(x)⁄_x is a good example -1.h-.   $\lim_{x \to 0^{+}} \frac{sin(x)}{x} = 0$, but f(0) is not defined.

f(x) = $\frac{x^{2}-4}{x-2}$. $\lim_{x \to 2}\frac{x^{2}-4}{x-2} = \lim_{x \to 2}\frac{(x-2)(x+2)}{x-2} = \lim_{x \to 2} (x+2) = 4,$ but f(2) is undefined.

$f(x) = \begin{cases} 2x + 1, &x<1\\\ 2, &x=1\\\ -x + 4, &x > 1 \end{cases}$

$\lim_{x \to 1^{+}} f(x) = 3 = \lim_{x \to 1^{-}} f(x), but \lim_{x \to 1} f(x) = 3 ≠ 2 = f(1)$

• f has an infinite discontinuity at a if $\lim_{x \to a^{+}} f(x) = \pm\infty$ or $\lim_{x \to a^{-}} f(x) = \pm\infty.$ Examples:

y = ¹⁄_x -1.f.- where $\lim_{x \to 0^{+}} \frac{1}{x} = \infty$ and $\lim_{x \to 0^{-}} \frac{1}{x} = -\infty$

y = ^(x+2)⁄_(x+1) where $\lim_{x \to -1^{+}} \frac{x+2}{x+1} = \infty$ and $\lim_{x \to -1^{-}} \frac{x+2}{x+1} = -\infty$

• A mixed discontinuity is a discontinuity that results when at least one of the one-sided limits does not exist. An example is y = sin(¹⁄_x) -1.i.-

Theorem. If a function f is differentiable at x = a, then f is continuous at x = a.

Proof. We need to check if $\lim_{x \to a} f(x) = f(a) ↔ \lim_{x \to a} f(x) - f(a) = 0$

$\lim_{x \to a} f(x) - f(a) = \lim_{x \to a} \frac {f(x) - f(a)}{x - a}(x-a) =$[By assumption, f is differentiable at x = a] f’(a)·0 = 0.

Bibliography

1. NPTEL-NOC IITM, Introduction to Galois Theory.
2. Algebra, Second Edition, by Michael Artin.
3. LibreTexts, Calculus. Abstract and Geometric Algebra, Abstract Algebra: Theory and Applications (Judson).
4. Field and Galois Theory, by Patrick Morandi. Springer.
5. Michael Penn, and MathMajor.
6. Contemporary Abstract Algebra, Joseph, A. Gallian.
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8. MIT OpenCourseWare 18.01 Single Variable Calculus, Fall 2007 and 18.02 Multivariable Calculus, Fall 2007.
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