Functional continuity

Let y = f(x) be defined in a certain neighborhood of x0, if the increment of the independent variable $\Delta x = x - x0$ approaches zero, the corresponding function value variation $\Delta y = f(x_0 + \Delta x) - f(x_0)$ also approaches zero, that $\lim _{\Delta x \to 0}\Delta y = 0$ is, the function y = f (x) is continuous at point x0.

How to understand this definition is very clear from the picture

Original address of the picture: https://www.wendangwang.com/doc/ce215e8c225ce3321899ebc7/2

If the function is continuous at x0 (left figure), then whether $\Delta x$ it is positive or negative (x changes to the right or left), the corresponding $x_0 + \Delta x$ function value will be infinitely close $f(x_0)$ , so the amount of change in the function value $Δy = 0$ . It can be seen from the figure that the curve is not broken.

If the function is discontinuous at x0 (as shown in the figure on the right), when $\Delta x$ < 0, it approaches x0 infinitely from the left side of x0, as can be seen from the figure $Δy = 0$ ; but $\Delta x$ when >0, it approaches x0 infinitely from the right side of x0, $\Delta and \neq$ . In this case, the function is continuous only on the left side and the curve is disconnected.

One-sided continuous definition of a function:

If the left limit of the function y = f(x) $\lim _{x \to x_{\bar{0}-}} = f(x_{0-})$ exists and is equal $f(x_0)$ , then f(x) is said to be continuous on the left of point x0;

If the right limit $\lim _{x \to x_{\bar{0}+}} = f(x_{0+})$ exists and is equal to $f(x_0)$ , f(x) is said to be right continuous at point x0.

A necessary and sufficient condition for a function to be continuous at a point is that it is both left continuous and right continuous at that point.

Interval continuity: If the function is continuous at every point in the open interval (a,b), then the function is continuous in (a,b). If the function is continuous in the closed interval [a,b], then the function is continuous in (a,b), and the left endpoint a is continuous to the right, and the right endpoint b is continuous to the left.

function discontinuity

Let the function y = f(x) be defined in some decentered neighborhood of point x0. If f(x) has one of the following three situations:

1. Not defined at x0;

2. There is a definition at x0, but the limit $\lim _{x \to x_0} f(x)$ does not exist;

3. There is definition at x0, the limit $\lim _{x \to x_0} f(x)$ exists, but $\lim _{x \to x_0} f(x) \neq f(x_0)$ ;

Then the function y = f(x) is discontinuous at x0, and x0 is called the discontinuity point of f(x). Let’s look at several different types of discontinuity points:

1. The function image of tanx is as follows

Take, $x = \pi / 2$ for example, that the function $x = \pi / 2$ is not defined at , this point is a discontinuity point of tanx, since $\lim _{x \to \pi/2} f(x) = \infty$ , therefore, this point is an infinite discontinuity point of tanx .

2. $y = sin\frac{1}{x}$ Image

This function is not defined at x = 0. On the left and right sides of the x = 0 point, the function value keeps vibrating between [-1,1], and the x = 0 point is the shock discontinuity point of the function .

3. $y = \frac{x^2 - 1}{x + 1}$ Image

This function has no definition at x = -1, but both the left and right limits exist at x = -1, and the left and right limits are equal to -2. If the point (-1,-2) is added, the function is continuous, and x = -1 is called the discontinuity point of the function .

4. $f(x) = \left \{ x -1 (x < 0); 0(x = 0); x + 1(x > 0) \right \}$

This function is defined at x = 0, the left and right limits exist, but the left and right limits are not equal, and the point x = 0 is called the jumping discontinuity point of the function .

There are generally two types of discontinuities:

The first type of discontinuity point (removable discontinuity point, jumping discontinuity point), x0 is the discontinuity point of f(x), the left limit and the right limit both exist (not necessarily equal).

The second type of discontinuity point (infinite discontinuity point, oscillation discontinuity point), x0 is the discontinuity point of f(x), and at least one of the left limit and the right limit does not exist.

reference

Lecture 9 Continuity of function and discontinuity of function- Know about

Function Continuity and Discontinuity

Functional continuity

function discontinuity

reference

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