Continuous, Differentiable and Differentiable

continuous

To determine whether a function is continuous at a certain point, you can use the following method:

1. Definition of function value: To determine the function$f(x)$ 在某点$x=a$If it is or not, demand is guaranteed$f\left(a\right)$existence, also being a function Existing$x=a$处处具繉。

2. The existence of left and right limits: A characteristic of a continuous function is that the left and right limits exist and are equal. If for function $f(x)$，${\lim }_{x\to a^{-}}f(x)$ 和${\lim }_{x\to a^{+}}f\left(x\right)$Existence, parallelism $f\left(a\right)$，Nano function exists< /span>$x=a$Continue.

3. Properties of limits: Function at a certain point$x=It\; is\; continuous\; at\; a$, which means that the limit and function value near this point are consistent. That is to say, ${\lim }_{x\to a}f(x)=f\left(a\right)$. This means that the arrival point is the near point$For\; any\; value\; of\; a$, the function value approaches $f(a)$。

If the above conditions are met, the function is at a certain point$x=a$ is continuous. If any of the conditions are not met, the function is not continuous at that point.

Satisfies 1, 2, does not satisfy 3

Let's look at an example:

Consider the function$f\left(x\right)=\{\begin{array}{ll}{x}^2,& \text{result}x\ne 2\\ 5,& \text{result}x=2\end{array}$

The definition of this function is clearly expressed in paragraphs $x=The\; situation\; at\; 2$. Let's check that this function is at $x=Continuous\; situation\; at\; 2$:

1. Definition of function value: The function is at $x=2$ 处SET义为$f(2)=5$, the function is defined at this point.

2. Limited existence:${\lim }_{x\to 2^{-}}f\left(x\right)={\lim }_{x\to 2^{-}}{x}^2=4$。${\lim }_{x\to 2^{+}}f\left(x\right)={\lim }_{x\to 2^{+}}{x}^2=4$. Both limits exist and are equal to $4$。

3. 极限的性质：${\lim }_{x\to 2}f\left(x\right)={\lim }_{x\to 2}{x}^2=4\ne f\left(2\right)=5$. Existence$x=The\; limit\; at\; 2$ is not equal to the function value.

Therefore, this function is at$x=2$ satisfies the definition of function value and the existence of left and right limits, but does not satisfy the properties of limits, so at $x=2$ is not continuous.

Satisfies 1, does not satisfy 2, 3

Consider the following function:

$f(x)=\{\begin{array}{ll}2x,& \text{result}x<1\\ x+3,& \text{result}x\ge 1\end{array}$

Let's check the continuity of this function at ( x = 1 ):

1. Definition of function value: The function is defined at ( x = 1 ) as$f(1)=1+3=4$, the function is defined at this point.

2. Limited existence:${\lim }_{x\to 1^{-}}f\left(x\right)={\lim }_{x\to 1^{-}}2x=2$。${\lim }_{x\to 1^{+}}f\left(x\right)={\lim }_{x\to 1^{+}}(x+3)=4$. These two limits exist, but are not equal.

3. Conclusive property：${\lim }_{x\to 1}f\left(x\right)$Absence, cause left and right极finite incompatibility, etc.

So, this function is at$x=1$ satisfies the definition of function value, but does not satisfy the existence of left and right limits and the properties of limits, so x = 1 x at $x=1$ is not continuous.

Satisfies 2, does not satisfy 1 and 3

Consider the following function:

$f\left(x\right)=\frac{\sin (x)}{x}$

The properties of this function at ( x = 0 ) are:

1. Definition of function value: The function is not well defined at ( x = 0 ) (because the denominator is zero), that is, ( f(0) ) does not exist.

2. The existence of left and right limits:${\lim }_{x\to 0^{-}}\frac{\sin \left(x\right)}{x}=1$。${\lim }_{x\to 0^{+}}\frac{\sin \left(x\right)}{x}=1$. Both limits exist and are equal.

3. Properties of limit: Since the function value is in$x=0$ is not defined, so the function value cannot be compared with the limit.

Therefore, this function satisfies the existence of left and right limits and the properties of limits, but does not satisfy the definition conditions of function values.

Differentiability and Differentiability

Differentiability and differentiability are two important concepts in calculus. They describe the smoothness and change of a function at a certain point.

1. Differentiability: For a single variable function, if the function has a unique tangent line (that is, the derivative exists) near a certain point, then the function at this point is Derivable. Differentiability means that the function is smooth at this point, with no sharp points or discontinuities. For multivariable functions, differentiability usually means that partial derivatives exist, and a linear function (tangent plane) can be used to approximately describe the change of the function near this point.

2. Continuously Differentiable: Differentiability is an extension of differentiability. It requires that the function is not only differentiable at a certain point, but also that the derivative is continuous near that point. . Differentiability means that the function is not only smooth near this point, but also the derivative changes continuously.

These two concepts are very important in describing the smoothness and rate of change of a function. Differentiability and differentiability allow us to understand the rate of change of a function at a point, which is critical for many problems such as finding extreme points in optimization problems, predicting the behavior of functions, solving differential equations, and for Modeling secondary problems in physics, engineering, and economics.

Differentiability and differentiability can be described in mathematical language as follows:

1. Differentiability (function of one variable): For a function of one variable$f\left(x\right)$, There is a certain point $x=a$, existential finite leader number $f^{\prime }\left(a\right)$, there are functions with different names. Point$x=It\; is\; derivable\; at\; a$. The mathematical expression is:

   ${\lim }_{h\to 0}\frac{f(a+h)-f(a)}{h}$

   If this limit exists, then the function is $x=a$处可导.

2. Differentiability (single variable function): If function$f\left(x\right)$A certain point$x=It\; is\; differentiable\; at\; a$, and the derivative$f^{\prime }(x)$在$x=a$处连续，な么function数在$x=a$It is possible.

For multivariable functions:

1. Differentiability (multivariable functions): For multivariable functions$f(x_1,x_2,\dots ,x_n)$，A certain point in the world $(x_1=a_1,x_2=a_2,\dots ,x_n=a_{n}The\; partial\; derivative\; at\; )$ exists, then the function is differentiable at this point.

2. Differentiability (multivariable function): If a multivariable function is differentiable at a certain point, and each partial derivative is continuous at that point, then the function at that point is Differentiable.

These concepts describe the smoothness and rate of change of a function at a certain point and are very important for analyzing the behavior and properties of functions.

Why is there so much emphasis on differentiability or differentiability in high school mathematics?

The emphasis on differentiability and differentiability in calculus stems from their widespread use in mathematics and the real world. These concepts provide us with the tools to understand the behavior and rate of change of functions and are critical for describing phenomena in nature, optimization problems, and modeling in fields such as physics and engineering.

Differentiability (for single-variable functions means differentiability, for multi-variable functions means the existence of partial derivatives) means that the function has a tangent line at a certain point, that is, the function is smooth at that point , no sudden jumps or discontinuities. This property allows us to calculate the slope or rate of change of a function at a certain point, which is essential forsolving extreme values, optimization problems, and solving differential equations important.

Differentiability is the generalization of differentiability, which means that the function not only has a tangent line at a certain point, but also can be approximately represented by a linear function near the point. This property allows us to use differentials to estimate the local changes of the function, and then apply it to mathematical tools such as Taylor expansion and least squares method.

The importance of these concepts is thatthey provide mathematical tools for dealing with changes and trends. These tools are used in the description of nature and the analysis of engineering problems. It has a very wide range of applications in solving problems and scientific research. Differentiability and differentiability in calculus provide us with important tools for understanding the behavior of functions, helping us to build models, predict phenomena, optimize problems, and make valuable predictions and decisions in many fields.