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

  • 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.

  • The Mean Value Theorem, Rolle's Theorem, and Monotonicity

    • The MVT states that for a function continuous on an interval, the mean value of the function on the interval is a value of the function.
    • More precisely, if a function $f$ is continuous on the closed interval $[a, b]$, where $a < b$, and differentiable on the open interval $(a, b)$, then there exists a point $c$ in $(a, b)$ such that
    • Rolle's Theorem states that if a real-valued function $f$ is continuous on a closed interval $[a, b]$, differentiable on the open interval $(a, b)$, and f(a) = f(b), then there exists a c in the open interval $(a, b)$ such that $f'(c)=0$.
    • For any function that is continuous on $[a, b]$ and differentiable on $(a, b)$ there exists some $c$ in the interval $(a, b)$ such that the secant joining the endpoints of the interval $[a, b]$ is parallel to the tangent at $c$.
    • Use the Mean Value Theorem and Rolle's Theorem to reach conclusions about points on continuous and differentiable functions

  • Functions of Several Variables

    • As we will see, multivariable functions may yield counter-intuitive results when applied to limits and continuity.
    • Unlike a single variable function $f(x)$, for which the limits and continuity of the function need to be checked as $x$ varies on a line ($x$-axis), multivariable functions have infinite number of paths approaching a single point.Likewise, the path taken to evaluate a derivative or integral should always be specified when multivariable functions are involved.
    • We have also studied theorems linking derivatives and integrals of single variable functions.
    • A scalar field shown as a function of $(x,y)$.
    • Extensions of concepts used for single variable functions may require caution.

  • Linear Approximation

    • A linear approximation is an approximation of a general function using a linear function.
    • In mathematics, a linear approximation is an approximation of a general function using a linear function (more precisely, an affine function).
    • Linear approximation is achieved by using Taylor's theorem to approximate the value of a function at a point.
    • Given a twice continuously differentiable function $f$ of one real variable, Taylor's theorem states that:
    • For example, given a differentiable function with real values, one can approximate for close to by the following formula:

  • 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

  • Limit of a Function

    • The limit of a function is a fundamental concept in calculus and analysis concerning the behavior of a function near a particular input.
    • The limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input.
    • Informally, a function $f$ assigns an output $f(x)$ to every input $x$.
    • In particular, the many definitions of continuity employ the limit: roughly, a function is continuous if all of its limits agree with the values of the function.
    • The limit as the function approaches $x_0$ from the left does not equal the limit as the function approaches $x_0$ from the right, so the limit of the function at $x_0$ does not exist.

  • 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.

  • Higher Derivatives

    • The second derivative, or second order derivative, is the derivative of the derivative of a function.
    • The derivative of the function may be denoted by $f'(x)$, and its double (or "second") derivative is denoted by $f''(x)$.
    • This is read as "$f$ double prime of $x$," or "the second derivative of $f(x)$. " Because the derivative of a function is defined as a function representing the slope of function, the double derivative is the function representing the slope of the first derivative function.
    • This can continue as long as the resulting derivative is itself differentiable, with the fourth derivative, the fifth derivative, and so on.
    • A function $f$ need not have a derivative—for example, if it is not continuous.