CONTINUITY

A function is said to be continuous in a given interval if there is no break in the graph of the function in the entire interval range. At x = a, the following conditions must be satisfied:

1. f is defined at x = a
2. The limit of f(x) as x approaches a exists, i.e., lim f(x) as x → a
3. f(a) = lim f(x) as x → a

DISCONTINUITY

A function is discontinuous at a point if there is a break, hole, jump, or vertical asymptote at that point. A function is discontinuous if:

1. It is undefined at a point
2. The limit does not exist
3. The limit exists but is not equal to the function value f(a)

SAMPLE PROBLEM

f(x) = x² − 4

Step 1: Check if f(2) exists
f(2) = 2² − 4 = 0
Defined

Step 2: Find the limit
Since it’s a polynomial, we substitute directly:
lim f(x) as x → 2 = 2² − 4 = 0

Step 3: Compare
lim f(x) as x → 2 = f(2)

Answer: The function is continuous at x = 2.

Graph Description

When we input the values into a graph, we see a smooth parabola that opens upward. The graph is shaped like a “U” and has its lowest point at (0, 4).

As x moves toward any value, including x = 2, the function values from the left and right approach the same number. For example, as x approaches 2, the graph approaches 0 from both sides, and the function value at 2 is also 0.

There are no jumps, holes, or breaks anywhere on the graph, which shows that the function is continuous for all real numbers.

SAMPLE PROBLEM: PIECEWISE FUNCTION

f(x) =
  {
    3, if x < 2
    7, if x ≥ 2
  }

Step 1: Left-hand limit
lim f(x) as x → 2⁻ = 3

Step 2: Right-hand limit
lim f(x) as x → 2⁺ = 7

Step 3: Compare
Since 3 ≠ 7,
the limit does not exist at x = 2.

Graph Description: Jump Discontinuity

Inputting the values into a graph, we can see that for x < 2, the graph stays at y = 3, while for x ≥ 2, the graph is y = 7.

At x = 2, there is an open circle at (2, 3) and a filled dot at (2, 7).

We can clearly see a sudden jump in the graph, which visually represents a jump discontinuity because the graph does not connect smoothly at that point.

APPLICATIONS OF CONTINUITY

Physics: Smooth motion of objects.
  When a car moves, it doesn’t suddenly disappear and reappear somewhere else. It moves smoothly from one spot to another. That smooth movement is known as continuity.

Engineering: Design and constructions.
  Engineers design roads, bridges, and roller coasters with smooth curves. If there were sudden breaks or jumps in the design, it would be dangerous.

Heart Rate Monitoring:
  Our heartbeat changes gradually when we exercise. Doctors study smooth patterns in heart rate. Sudden irregular jumps may signal a problem.

APPLICATIONS OF DISCONTINUITY

Switching Circuits / Electronics:
  Turning a device ON/OFF causes current to change abruptly instead of gradually.

Stock Market:
  Sometimes stock prices jump suddenly due to big news or events. This sudden change is a discontinuity in the price chart.

Tax Brackets:
  When income passes a certain limit, the tax rate jumps suddenly. This abrupt change in rate is a real-life example of a discontinuity.

Removable Discontinuity

A removable discontinuity happens when a function has a hole at a point.
This occurs if the limit exists at that point but the function value is missing or does not match the limit.

Essential Discontinuity

An essential discontinuity occurs when the limit of the function does not exist (DNE) as it approaches a point.
This breaks the second condition for continuity: the limit of f(x) as x → a must exist.

Applications of Removable Discontinuity

Mathematical Models / Simulations:
  Some formulas are undefined at a single point (like dividing by zero). Redefining the value at that point fixes the “hole” in the model.

Engineering:
  When designing curves (roads, tracks, ramps), a tiny missing point or undefined spot can create a hole. Fixing that spot makes the curve smooth again.

Animation / Video Games:
  A character might jump slightly or a frame might be missing in an animation. Adding the missing frame “fills the hole” just like removing a removable discontinuity.

Applications of Essential Discontinuity

Physics:
  Modeling phenomena that have sudden spikes or singularities, like electric field strength near a point charge.

Engineering:
  Designing systems where abrupt changes occur, such as switching circuits that jump from one current to another instantly.

Finance / Economics:
  Representing sudden jumps in stock prices, interest rates, or other financial indicators where the values change abruptly at certain points.

Example: Determining Continuity at x = 2

Consider the function defined piecewise as:

f(x) =
  {
    (x − 2)², if x < 2
    (x − 2)² + 3, if x ≥ 2
  }

Step 1: Left-hand limit
  Use the first piece (x < 2) and plug in x = 2:
  f(x) = (2 − 2)² = 0
  Left-hand limit = 0

Step 2: Right-hand limit
  Use the second piece (x ≥ 2) and plug in x = 2:
  f(x) = (2 − 2)² + 3 = 3
  Right-hand limit = 3

Conclusion:
  Since the left-hand limit ≠ right-hand limit, the function is discontinuous at x = 2.

Graph Description: Jump Discontinuity

Inputting the function into a graph, we can see that the value from the right as x approaches 2 is 3, while from the left it is approaching 0.
We can clearly see in the graph the visual representation of the jump discontinuity as the function jumps from 0 to 3.
Open and filled circles at x = 2 help highlight the jump:
  • Hollow circle at (2,0) for the left-hand value.
  • Filled circle at (2,3) for the right-hand value.

Example: Determine Continuity at x = 4

Consider the function:

f(x) = 1 / (x − 4)

Step 1: Right-hand limit
  For x > 4, x - 4 > 0.
  As we input values just to the right of 4, f(x) = 1 / (x - 4) gets larger and larger.
  Meaning, it is approaching positive infinity.

Step 2: Left-hand limit
  For x < 4, x - 4 < 0.
  As we input values just to the left of 4, f(x) = 1 / (x - 4) becomes increasingly negative.
  Meaning, it is approaching negative infinity.

Conclusion:
  Since the left-hand limit → −∞ and the right-hand limit → ∞, the function is infinitely discontinuous at x = 4.

Graph Description: Infinite (Essential) Discontinuity

In the graph, as we input the function, we can see that it is approaching infinity from both the left and right sides.
There is a clear vertical asymptote at x = 4.
This visually represents the infinite discontinuity as the function shoots up and down without bound near x = 4.

Applications of Jump (Essential) Discontinuity

Digital Logic and Computing:
  A computer bit is either 0 or 1. When a signal changes from "off" to "on," we model it as a jump. There is no "0.5" state in binary logic.

Income Tax Brackets:
  Your marginal tax rate jumps to a higher percentage when income crosses certain thresholds.

Quantum Jumps:
  In physics, electrons in an atom don't "slide" between energy levels. They disappear from one orbit and reappear in another instantly, illustrating a jump discontinuity in energy levels.

Applications of Infinite (Essential) Discontinuity

Radio Frequency Interference:
  If you move a microphone close to a speaker, you will get a feedback loop. The screech is an essential discontinuity in the sound wave.

Economics and Finance:
  Infinite growth of an investment or extreme market crashes represent sudden, unbounded changes, modeled as infinite discontinuities.

Physics and Engineering:
  Electric field strength near a point charge or gravitational potential near a massive object can change abruptly, illustrating infinite discontinuities.

Open Interval (a, b):
  A function is continuous on (a, b) if f(c) = lim x→c f(x) for every c in (a, b).

Closed Interval [a, b]:
  A function is continuous on [a, b] if:
    • It is continuous on the open interval (a, b).
    • It is continuous from the right at a: lim x→a⁺ f(x) = f(a).
    • It is continuous from the left at b: lim x→b⁻ f(x) = f(b).

Example: Interval of Continuity for a Polynomial

Determine the interval over which the function f(x) = x² − 2x + 4 is continuous.

Since the given function is a polynomial, it is defined for all real numbers.
A polynomial function is said to be continuous on the interval (-∞, +∞).

Example: Interval of Continuity for a Rational Function

Determine the interval over which the function f(x) = (x − 2) / (x + 4) is continuous.

Since the given function is a rational function, we must check where the denominator = 0.
  x + 4 = 0 → x = −4
At x = −4, the function is undefined.

Therefore, the function is continuous everywhere except at x = −4.
The interval of continuity is:
  (-∞, −4) ∪ (−4, +∞)