continuity

Examples of continuity in the following topics:

• Continuity

  • 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.
  • In fact, a dictum of classical physics states that in nature everything is continuous.
  • The function $f$ is said to be continuous if it is continuous at every point of its domain.

• Limits and Continuity

  • Continuity in each argument does not imply multivariate continuity.
  • 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 $f$.
  • Similarly, all $f_x$ are continuous as $f$ is symmetric with regards to $x$ and $y$.
  • Continuity in single-variable function as shown is rather obvious.
  • Describe the relationship between the multivariate continuity and the continuity in each argument

• Intermediate Value Theorem

  • 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 $2$.
  • Therefore, since it is continuous, there must be at least one point where $x$ is $0$.
  • Use the intermediate value theorem to determine whether a point exists on a continuous function

• Physics and Engeineering: Center of Mass

  • For a continuous mass distribution, the position of center of mass is given as $\mathbf R = \frac 1M \int_V\rho(\mathbf{r}) \mathbf{r} dV$ .
  • If the mass distribution is continuous with respect to the density, $\rho (r)$, within a volume, $V$, then the integral of the weighted position coordinates of the points in this volume relative to the center of mass, $\mathbf{R}$, is zero, that is:
  • If a continuous mass distribution has uniform density, which means $\rho$ is constant, then the center of mass is the same as the centroid of the volume.
  • COM can be defined for both discrete and continuous systems.

• 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

• Second-Order Linear Equations

  • A second-order linear differential equation has the form $\frac{d^2 y}{dt^2} + A_1(t)\frac{dy}{dt} + A_2(t)y = f(t)$, where $A_1(t)$, $A_2(t)$, and $f(t)$ are continuous functions.
  • where $A_1(t)$, $A_2(t)$, and $f(t)$ are continuous functions.

• Probability

  • 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:

• The Fundamental Theorem of Calculus

  • This part of the theorem is also important because it guarantees the existence of antiderivatives for continuous functions.
  • Let $f$ be a continuous real-valued function defined on a closed interval $[a,b]$.
  • Now, $F$ is continuous on $[a,b]$, differentiable on the open interval $( a,b )$, and $F'(x) = f(x)$ for all $x$ in $( a,b )$.

• Higher Derivatives

  • 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.

• 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.