Lecture 12: Continuity of functions

12.1 Assessment Content

• Content
• 1. The concept of continuity;
• 2. Breaks and their classification;
• 3. Operations and properties of continuity;
• 4. Properties of continuous functions on closed intervals;
• Question type
• Question Type 1: Discuss the types of function continuity and discontinuity;
• Problem Type 2: Proof problems with properties of continuous functions on closed intervals;

12.2 The Concept of Continuity

#####Definition: Continuity

description:

Definition 1: $y=f(x)$ is continuous at $x_0$ if $\lim_{\Delta x\to0}[f(x_0+\Delta x)-f(x_0)]=0$.

Definition 2: $y=f(x)$ is continuous at $x_0$ if $\lim_{x\to x_0}f(x)=f(x_0)$.

Explanation

• Equivalent form: $y=f(x)$ is continuous at $x_0$ iff $\lim_{x\to x_0}f(x)=f(x_0)$.
• Left and right continuous:
• 1. Left continuous: $\lim_{x\to x^-_0}f(x)=f(x_0)$
• 2. Right continuous: $\lim_{x\to x^+_0}f(x)=f(x_0)$
• $f(x)$ Sufficient and necessary conditions for continuity: $f(x)$Left continuous and right continuous,left and right continuous are equal;

Inner and Outer Continuities

• Inner continuous: every point in $(a,b)$ is continuous;
• Outer continuous: Every point in $[a,b]$ is continuous, and point a is continuous to the right and point b to the left;

Continuous within the range

• A function is continuous within an interval = all points on the interval are continuous =The left and right limits of all points exist and are equal, and the function value equals this point;

Question Type: Discussthecontinuityoffunctionsandtypesofdiscontinuities

PART 1: Problem-solving methods

• When discussing the continuity of functions, it is necessary to clearly describe which points are discontinuous and why they are continuous on other intervals;

PART 2: Typical Example Problems

Example Problem: $\text{ Known }f(x)=\begin{cases}(\cos x)^{1/x^2},x\neq0,\\a,&x=0&\end{cases}$, then continue at x=0, then find a;

• Analysis
• Because it is continuous at x=0, the limit value of this point exists and is equal;
• Analysis
• $\lim_{x\to0}f(x)=\lim_{x\to0}(nx)^{\frac12}=\lim_{x\to0}[1+(ax-1)]^{\frac1{x^2}}=e^{-\frac12}=f(0)=0,$
• Question Type: #

Example Question: $f(x)=\lim_{n\to\infty}\frac{1+x}{1+x^{{2n}}}$, discuss the breakpoint of the function;

• Analysis
• Analysis
• First, segment it to get: $f(x)=\begin{cases}1+x&|x|<1\\0&|x|>1\\1&x=1\\0&x=-1\end{cases}$
• 0 points continuous, 1 point discontinuous;
• Therefore: there exists a discontinuity point x=1;
• Question Type: #