Continuity Core

The Three-Part Definition of Continuity

UNIT: LIMITS & CONTINUITY

MOD: 01

The "Unbroken" Challenge

We often say a function is continuous if you can draw its graph without lifting your pencil.

"Is this intuitive definition enough for a structural engineer? Or a computer scientist?"

Can you draw a curve where...

The limit exists at \(x = c\)
The function is defined at \(x = c\)
BUT there is still a hole?

The Triple Check

A function \(f(x)\) is continuous at a point \(c\) if and only if these three criteria are met:

1

\(f(c)\) is defined

The function exists at that spot

2

\(\displaystyle \lim_{x \to c} f(x)\) exists

The curve approaches the same target

3

\(\displaystyle \lim_{x \to c} f(x) = f(c)\)

The curve and target are the same

Failure 01: The Void

When \(f(c)\) is undefined

Algebraically, this usually happens when you divide by zero or take the square root of a negative number.

\[f(x) = \frac{x^2 - 4}{x - 2}\]

At \(x = 2\), the function does not exist.

\(x = c\)

Hole (Removable Discontinuity)

Failure 02: The Disconnect

When \(\lim_{x \to c} f(x)\) DNE

This happens if the left-hand limit and right-hand limit disagree.

\[\lim_{x \to c^-} f(x) \neq \lim_{x \to c^+} f(x)\]

The curve "jumps" or goes to infinity.

Jump Discontinuity

Failure 03: The Mismatch

When Limit \(\neq\) Value

The point is defined, the limit exists, but the point is in the wrong place.

\(\displaystyle \lim_{x \to 3} f(x) = 5\)

\(f(3) = 10\)

DISCONTINUOUS

Point Discontinuity (The "Fly-Away" Point)

Standard of Proof

In Calculus, writing "it's unbroken" is worth zero points.

Sample Response Structure:

"The function \(f(x)\) is discontinuous at \(x=c\) because although \(f(c)\) is defined as \(L_1\) and the \(\lim_{x \to c} f(x)\) exists as \(L_2\), the two values are not equal (\(L_1 \neq L_2\))."

3 Steps

Verify

Specific Values

Calculated

Conclusion

Final Verdict

Rapid Analysis

Test the continuity of \(g(x)\) at \(x = -1\):

\[g(x) = \begin{cases} 2x + 5 & x \leq -1 \\ x^2 + 2 & x > -1 \end{cases}\]

Step 1

Is \(g(-1)\) defined?

Step 2

Does the limit exist?

Step 3

Are they equal?

Continuity Criteria and Existence Theorems

The Three Part Definition of Continuity

Classifying Discontinuities

Continuity on Intervals and Composites

The Intermediate Value Theorem

Squeeze Theorem and Special Limits

Continuity Core Slides

Triple Threat Checklist Worksheet

Continuity Facilitation Guide

Break Point Slides

Broken Tracks Activity Worksheet

Interval Integrity Slides

Continuity Toolbox Reference Sheet

Chain Reaction Worksheet

Existence Proof Slides

Root Hunter Worksheet

Squeeze Play Slides

Trig Limit Mastery Worksheet

Similar Sequences

Math Marathon TSIA2 Prep

Penne Reed

Decimal Dimensions

Calculating Chance

Adulting Math Bootcamp

Hannah Milford

Mathematical Marvels

Math Mastery

Brenda Coombs

Real World Math

V Lundgren

Trigonometry Waves

hpagan

Trig Foundation Toolkit

Statistiques Exploratoires

Hugo Pierrard

Radical Relationships

Calculus Foundations

Transformation Surgeons

Sigma Summit

Exponential Strategy Masterclass

Manipulative Math Mastery

Blueprint Fractions and Ratios

Fraction Blueprint Workshop

Spatial Reasoning and Geometric Modeling

Curvilinear Coordinates and Vector Analysis

Mastering Logic Structures

Rigorous Proofs in Analysis