Continuity

Much like the concept of limits, continuity is generally one of the underlying building blocks of analytical relations and numerical methods used in engineering. Engineers like to deal with smooth, continuous functions, since they behave in a way that makes it easier to find additional information about the physics of the problem, such as identifying rates of change. However, engineers often interact with physical phenomena that are discontinuous by nature! Understanding continuity will allow engineering students to better understand the challenges and limitations of working with a specific physics model.

Engineering Context:

• MAE:
  When evaluating the force on any structural member, such as a beam, the force is often spread across the length of the member. For example, a cantilevered beam (a beam that is fixed on one end and free on the other) will experience a load that increases in a linear fashion along the length of the beam. An analysis of the forces and moments acting on the beam often begins by representing the effects of a load through a shear force and bending moment diagram. The internal bending moment of the beam when expressed as a function of the length of the beam is equal to the integral of the shear force up to a given point:

  $M\left(x\right)={\int }_0^{x}V\left(x\right)dx$

  Umani, Kingsley. “Shear Force and Bending Moment diagram in the horizontal plane.” ResearchGate, August 2020, https://www.researchgate.net/figure/Shear-Force-and-Bending-Moment-diagram-in-the-horizontal-planef ig4343978691.

  The force and moment diagrams are not always continuous; thus, understanding the principles of continuity is necessary for empirical and numerical analysis. In a shear force diagram, the force is continuous along the beam so long as no external force is applied and no change is made in the beam loading. With an external force applied, the magnitude of the shear force acting on the beam jumps by an equivalent amount, which creates a discontinuity. When creating the moment diagram, continuity is maintained between applied forces. Any discontinuity in this diagram, similar to the shear force diagram, indicates an applied moment at that point along the beam. The magnitude of the jump is equal to the magnitude of the applied moment.

• ECE:
  Capacitors follow a general principle of continuity, meaning that the voltage across a capacitor cannot change instantaneously unless it is under the influence of an infinite current. Similarly, in the absence of an infinite voltage, the current through an inductor cannot change instantaneously. Essentially, this means that a graph of voltage across a capacitor or current through an inductor will always be continuous. Voltage in a capacitor is modeled by the equation

  $v=\frac{q}{C}$

  which tells us that voltage is dependent on how much charge, q, is stored within the capacitor. The charge stored within the capacitor is associated with definite particles (electrons), which cannot spontaneously appear or disappear. Therefore, assuming that the voltage cannot spontaneously jump up or down in value is reasonable. Understanding that spontaneous changes in voltage cannot occur is helpful in the design of circuits, since these worst-case scenarios need not be integrated into the design.

• BE: In engineering applications, it’s quite likely that you will encounter various types of discontinuous functions. Piecewise functions are especially common when representing physical phenomena. For example, say a biological engineer is modeling bacterial growth, and they want to know how sudden changes in the bacterial environment might affect the bacteria’s growth rate. They culture the bacteria in a nutrient-poor environment until time t, at which point they suddenly add nutrient-rich media to the culture. This causes a discontinuous jump in the growth rate.

• CEE:

  

  Figure 1: Shear Force and Bending Moment Diagram

  The internal forces on beams and frames follow a pattern of continuity where shear force and bending moment are continuous except at point loads, applied moments, and joints. Just like with Mechanical and Aerospace Engineers, the analysis of these structures is influenced directly by looking at shear and bending moment diagrams. At these three kinds of points, shear force and bending moment can undergo a discontinuous jump. This is an important concept to understand when designing structures in order to ensure that a given structure will be able to withstand the various loads applied to it, especially when the loads cause a discontinuous jump in either shear or bending moment. These step changes in force or moment require additional considerations in material mechanics, such as understanding yielding and buckling.

The Essentials

Continuity of a function can either be defined at a point or along some interval in the domain of the function. Both will be detailed here, along with two theorems that can be employed once we know if a function is continuous.

Types of Discontinuities

There are three types of discontinuities: removable discontinuities, jump discontinuities, and infinite discontinuities.

Removable Discontinuity

If the function $f\left(x\right)$ is discontinuous at a, it has a removable discontinuity if:

$\underset{x\to a}{lim}f\left(x\right)$ exists, but $f\left(a\right)$ is not defined

Jump Discontinuity

If the function $f\left(x\right)$ is discontinuous at a, it has a jump discontinuity if:

$\underset{x\leftarrow a}{lim}f\left(x\right)\text{and}\underset{x\to a^{+}}{lim}f\left(x\right)\text{both exist, but}\underset{x\to a^{-}}{lim}f\left(x\right)\ne \underset{x\to a^{+}}{lim}f\left(x\right)$

Infinite Discontinuity

If the function $f\left(x\right)$ is discontinuous at a, it has an infinite discontinuity if:

$\underset{x\leftarrow a}{lim}f\left(x\right)=\pm \infty \text{ or}\underset{x\to a}{lim}f\left(x\right)=\pm \infty$

Figure 2: A removable discontinuity.

Figure 3: A jump discontinuity.

Figure 4: An infinite discontinuity.

Continuity over an Interval

A function $f\left(x\right)$ is continuous from the right at a if $\underset{x\to a+}{lim}f\left(x\right)=f\left(a\right)$ .

A function $f\left(x\right)$ is continuous from the left at a if $\underset{x\to a-}{lim}f\left(x\right)=f\left(a\right)$ .

Composite Function Theorem

Theorem 1. If $f\left(x\right)$ is continuous at L and $\underset{x\to a}{lim}g\left(x\right)=L$ , then

$\underset{x\to a}{lim}f\left(g\right(x\left)\right)=f\left(\underset{x\to a}{lim}g\left(x\right)\right)=f\left(L\right)$ .

Intermediate Value Theorem (IVT)

Theorem 2. If f is continuous over the closed, bounded interval [a, b] and z is any real number between $f\left(a\right)$ and $f\left(b\right)$ , then there is a number c in [a, b] that satisfies $f\left(c\right)=z$ .

A visual representation of this theorem is given in Fig. 5.

Figure 5: Representation of the intermediate value theorem.

A Deeper Dive

If a function is continuous, it is often said it can be drawn without lifting the pencil off the paper. It is important to be able to determine whether a function is continuous or not, since many of the laws and rules associated with limits are dependent on a function being continuous on its domain or over an interval. Fortunately, many physical relationships can be described using a polynomial and all polynomials are continuous at every point on their domains. This allows us to evaluate limits of polynomials, and means that if p(x) is a polynomial, then:

Using this knowledge we can apply the same theorem to rational functions and state that if p(x) and q(x) are polynomials then

Trigonometric functions are also continuous over their entire domain, and many physical relationships are trigonometric in nature. When computing the limit and evaluating continuity of a trigonomentric function, it is important to use the Composite Function Theorem, which is given in Theorem 1.

Practice

More Resources

• MrMath: Continuity External Link Icon
• Kahn Academy: Continuity External Link Icon
• Nagwa: Continuity of Functions External Link Icon