Precalculus Concepts: Continuous vs. Discontinuous
Read this topicExplanations 4
- There are formal ways to discuss and define continuous functions, but here is a "lazy" definition of continuous: Continuous functions are ones which you could, in principle, dra…
By Ishwar Karthik, Moor Xu, Ruiran Xun, Joshua Siktar, Caroline K
- An increasing or decreasing function that is defined on an interval and does not skip any intermediate values is continuous. This property can be taken as the definition of cont…
By Michael Livshits
- The common "draw without lifting a pencil" definition only applies to functions on certain domains, typically an open, closed, or clopen interval of real numbers for functions f…
By Yohan Piggle
- (Video) Where is a Function Continuous? by MySecretMathTutor Here's a video by MySecretMathTutor explaining the difference between continuous and discontinuous graphs.
By Alex Federspiel
Problems 21
- Which of the following functions are continuous?
- Let f(x)={cx(x−3)x−3x≠3c+10x=3f(x) = \begin{cases} \frac{cx(x-3)}{x-3} & x \not= 3 \\ c + 10 & x = 3 \end{cases}f(x)={x−3cx(x−3)c+10x=3x=3 for some real number ccc. If fff …
- Consider the piecewise linear function: f(x)={3x+10x≤−2ax+b−2<x<4x−12x≥4f(x) = \begin{cases} 3x+10 & x \le -2 \\ ax + b & -2 < x < 4 \\ x - 12 & x \ge 4 \end{cases}f(x)=⎩⎨⎧3x+1…
- Let fff and ggg be continuous functions and ccc be a constant, and define the function hhh as follows: h(x)={f(x)x≤cg(x)x>ch(x) = \begin{cases} f(x) & x \le c \\ g(x) & x > c \e…
- Select the intervals on which this graph is continuous.
- Select the intervals on which this graph is continuous.
- Select the intervals on which this graph is continuous.
- Is the following function continuous or discontinuous? y=x−3\\y=x^{-3}y=x−3
- Is y=ln(x)+2xy=\ln(x)+2^xy=ln(x)+2x continuous or discontinuous on its domain?
- True or False? Rational functions are discontinuous over (−∞,∞)(-\infty,\infty)(−∞,∞).
- At what x−x-x−value(s) is f(x)=x−33x2−5x−12f(x)=\frac{x-3}{3x^2-5x-12}f(x)=3x2−5x−12x−3 discontinuous?
- Which of the following functions are continuous over (−∞,∞)(-\infty, \infty)(−∞,∞)? Select all correct answers.
- f(x)={2x2+3x+1x<03x+4+cx≥0f(x)= \begin{cases} 2x^2+3x+1 & x<0 \\ \sqrt{3x+4}+c & x\ge 0 \end{cases}f(x)={2x2+3x+13x+4+cx<0x≥0 The given function is continuous. Find the value…
- What xxx-value has discontinuity in the following function: f(x)=x2−x−2x−2f(x)=\frac{x^{2}-x-2}{x-2}f(x)=x−2x2−x−2
- In the following function, where are the points of discontinuity? f(x)=x+3x3+3x2−x−3f(x)=\frac{x+3}{x^3+3x^2-x-3}f(x)=x3+3x2−x−3x+3
- True or False? All one-to-one functions are continuous over (−∞,∞)(-\infty,\infty)(−∞,∞).
- If f(x)=3x2−3f(x)=3x^2-3f(x)=3x2−3 and g(x)=x+1g(x)= x+1g(x)=x+1, which of the following are continuous over all real numbers?
- A lion is walking around hunting for their prey. They walk in a wavy line for 242424 feet. All of a sudden, they jump 222 feet and land on the ground again just in the right pla…
- In the following function, where are the points of discontinuity on the domain (−∞,∞)(-\infty, \infty)(−∞,∞)? f(x)=2x2+3x2+17f(x)=\frac{2x^2+3}{x^2+17}f(x)=x2+172x2+3
- Where are the points of discontinuity in the following function? f(x)=(x2−x−12)(x2−5x+4)f(x)=\frac{(x^{2}-x-12)}{(x^{2}-5x+4)}f(x)=(x2−5x+4)(x2−x−12)
- True or False Continuous functions that have both increasing and decreasing sections cannot be one-to-one.