continuous function

(noun)

a function whose value changes only slightly when its input changes slightly

Related Terms

Examples of continuous function in the following topics:

  • Continuity

    • A continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output.
    • A continuous function with a continuous inverse function is called "bicontinuous."
    • Continuity of functions is one of the core concepts of topology.
    • This function is continuous.
    • The function $f$ is said to be continuous if it is continuous at every point of its domain.

  • Intermediate Value Theorem

    • For a real-valued continuous function $f$ on the interval $[a,b]$ and a number $u$ between $f(a)$ and $f(b)$, there is a $c \in [a,b]$ such that $f(c)=u$.
    • Since $0$ is less than $1.6$, and the function is continuous on the interval, there must be a solution between $1$ and $5$.
    • Version 3: Suppose that $I$ is an interval $[a, b]$ in the real numbers $\mathbb{R}$ and that $f : I \to R$ is a continuous function.
    • This captures an intuitive property of continuous functions: given $f$ continuous on $[1, 2]$, if $f(1) = 3$ and $f(2) = 5$, then $f$ must take the value $4$ somewhere between $1$and $2$.
    • Use the intermediate value theorem to determine whether a point exists on a continuous function

  • The Intermediate Value Theorem

    • For each value between the bounds of a continuous function, there is at least one point where the function maps to that value.
    • The function is defined for all real numbers x ≠ −2 and is continuous at every such point.
    • There is no continuous function F: R → R that agrees with f(x) for all x ≠ −2.
    • In plotting a continuous and smooth function between two points, all points on the function between the extremes are described and predicted by the Intermediate Value Theorem.
    • A graph of a rational function, .

  • Five-Part Rondo

    • Hybrid 1 combines the antecedent phrase (typically associated with the period) with the continuation phrase (typically associated with the sentence).
    • This results in a complete presentation–continuation–cadential function progression in the antecedent phrase followed by an incomplete continuation–cadential function progression.
    • On the large scale, the antecedent phrase functions like a big presentation function zone (like the presentation phrase does).
    • The CBI expresses presentation function, followed by a continuation phrase that expresses continuation and cadential functions.
    • Continuation and cadential function do not appear until the last contrasting idea (CI).

  • Limits and Continuity

    • A study of limits and continuity in multivariable calculus yields counter-intuitive results not demonstrated by single-variable functions.
    • A study of limits and continuity in multivariable calculus yields many counter-intuitive results not demonstrated by single-variable functions .
    • For instance, in the case of a real-valued function with two real-valued parameters, $f(x,y)$, continuity of $f$ in $x$ for fixed $y$ and continuity of $f$ in $y$ for fixed $x$ does not imply continuity of $f$.
    • Continuity in single-variable function as shown is rather obvious.
    • However, continuity in multivariable functions yields many counter-intuitive results.

  • Continuous Probability Distributions

    • A continuous probability distribution is a probability distribution that has a probability density function.
    • Mathematicians also call such a distribution "absolutely continuous," since its cumulative distribution function is absolutely continuous with respect to the Lebesgue measure $\lambda$.
    • The definition states that a continuous probability distribution must possess a density; or equivalently, its cumulative distribution function be absolutely continuous.
    • This requirement is stronger than simple continuity of the cumulative distribution function, and there is a special class of distributions—singular distributions, which are neither continuous nor discrete nor a mixture of those.
    • In theory, a probability density function is a function that describes the relative likelihood for a random variable to take on a given value.

  • Continuous Sampling Distributions

    • Now we will consider sampling distributions when the population distribution is continuous.
    • Note that although this distribution is not really continuous, it is close enough to be considered continuous for practical purposes.
    • Moreover, in continuous distributions, the probability of obtaining any single value is zero.
    • A probability density function, or density of a continuous random variable, is a function that describes the relative likelihood for this random variable to take on a given value.
    • Boxplot and probability density function of a normal distribution $N(0, 2)$.

  • Probability

    • Here, we will learn what probability distribution function is and how it functions with regard to integration.
    • In probability theory, a probability density function (pdf), or density of a continuous random variable, is a function that describes the relative likelihood for this random variable to take on a given value.
    • A probability density function is most commonly associated with absolutely continuous univariate distributions.
    • For a continuous random variable $X$, the probability of $X$ to be in a range $[a,b]$ is given as:
    • Apply the ideas of integration to probability functions used in statistics

  • Piecewise Functions

    • Piecewise functions are defined using the common functional notation, where the body of the function is an array of functions and associated intervals.  
    • The domain of the function starts at negative infinity and continues through each piece, without any gaps, to positive infinity.  
    • Since there is an closed AND open dot at $x=1$ the function is continuous there.  
    • When $x=2$, the function is also continuous.  
    • The range begins at the lowest $y$-value, $y=0$ and is continuous through positive infinity.  

  • Derivatives of Exponential Functions

    • The derivative of the exponential function is equal to the value of the function.
    • Functions of the form $ce^x$ for constant $c$ are the only functions with this property.
    • The rate of increase of the function at $x$ is equal to the value of the function at $x$.
    • If a variable's growth or decay rate is proportional to its size—as is the case in unlimited population growth, continuously compounded interest, or radioactive decay—then the variable can be written as a constant times an exponential function of time.
    • Graph of the exponential function illustrating that its derivative is equal to the value of the function.