User:SamHB/Continuity

<math>x+3=7</math> <math>x=?</math>

This article/section deals with mathematical concepts appropriate for a student in early high school.

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Put simply, a mathematical function is continuous if its graph can be drawn without lifting the pen from the paper. In the figures below, the graph on the left is of a continuous function; the graph on the right is not.

The function on the left is:

<math>f(x) = x^3 - 3x^2 + 2x + 1\,</math>

The function on the right is:

<math>f(x) = x^3 - 3x^2 + 2x + 1\,</math> for x <math>\le</math> 2

<math>f(x) = x^3 - 3x^2 + 2x - 1\,</math> for x <math>>\,</math> 2

More precise definition

<math>\frac{d}{dx} \sin x=?\,</math>

This article/section deals with mathematical concepts appropriate for late high school or early college.

In the language of calculus, there is a more precise definition in terms of limits—f(x) is continuous at a point X if the limit of f(x) as x approaches X is equal to the function value at that point, f(X):

<math>\lim_{x\to X}f(x) = f(X)\,</math>

From the definition of a limit, this means

For every ε > 0, there is a δ > 0 such that, whenever <math>0 < |x-X| < \delta, |f(x)-f(X)| < \varepsilon\,</math>

In the example in the picture on the right, the function is not contiuous at x=2. We have f(2) = 1, but it isn't continuous there. To see this, let ε=1. We would need to find δ such that

whenever <math>0 < |x-2| < \delta, |f(x)-1| < 1\,</math>

But for all x with 2 < x < 2.1, we have

f(x)-1 < -1.769

so

|f(x)-1| < 1.769

In topology

<math>\pi_1(S^1)=?\,</math>

This article/section deals with mathematical concepts appropriate for a student in late university or graduate level.

In the language of topology, there is an extremely simple and elegant formulation of continuity. In topology, all questions ultimately refer to open sets. A function is continuous if the inverse image of every open set is open. It is continuous at a point X if the inverse image of every open set containing X is open.