Mathematics
Grade 12
15 min
Determine one-sided continuity using graphs
Determine one-sided continuity using graphs
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Introduction & Learning Objectives
Learning Objectives
Define continuity from the left and continuity from the right.
Visually identify the left-hand and right-hand limits of a function at a specific point on a graph.
Determine if a function is continuous from the left at a given point by analyzing its graph.
Determine if a function is continuous from the right at a given point by analyzing its graph.
Explain the relationship between one-sided continuity and overall continuity at a point.
Analyze graphs of piecewise functions and functions on closed intervals to determine one-sided continuity at critical points and endpoints.
How does a company calculate shipping costs that suddenly jump at 1kg, 5kg, and 10kg? 📦 The math behind these 'jumps' is perfectly described by one-sided continuity!
In this...
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Key Concepts & Vocabulary
TermDefinitionExample
One-Sided Limit (from the right)The value that a function f(x) approaches as x gets infinitesimally close to a number 'c' from the positive side (values greater than c).For a graph where the curve approaches y=5 as you trace it from the right towards x=2, we write lim_{x→2⁺} f(x) = 5.
One-Sided Limit (from the left)The value that a function f(x) approaches as x gets infinitesimally close to a number 'c' from the negative side (values less than c).For a graph where the curve approaches y=3 as you trace it from the left towards x=2, we write lim_{x→2⁻} f(x) = 3.
Continuity from the RightA function f(x) is continuous from the right at x=c if the function's value at c is equal to the limit as x approaches c from the right. Both the limit and the...
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Core Formulas
Condition for Continuity from the Right
A function f is continuous from the right at x=c if \lim_{x \to c^+} f(x) = f(c)
Use this to test for continuity from the right. First, find the value the graph approaches as you trace it from the right side towards x=c. Then, find the y-value of the solid dot at x=c. If these two values are the same, it is continuous from the right.
Condition for Continuity from the Left
A function f is continuous from the left at x=c if \lim_{x \to c^-} f(x) = f(c)
Use this to test for continuity from the left. Find the value the graph approaches as you trace it from the left side towards x=c. Compare this to the y-value of the solid dot at x=c. If they match, it is continuous from the left.
Condition for Overall Continuity
A function f is co...
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Challenging
You are given that a function f(x) is continuous from the left at x=a, but it is NOT continuous at x=a. What must be true?
A.lim_{x -> a^-} f(x) does not exist.
B.f(a) is undefined.
C.f(x) is not continuous from the right at x=a.
D.lim_{x -> a^+} f(x) = f(a).
Challenging
On the graph of f(x), at x=c, the left-hand limit is L and the right-hand limit is M, where L is not equal to M. If f(c) is redefined to be equal to M, what is the effect on the one-sided continuity at x=c?
A.The function becomes continuous from the left.
B.The function becomes continuous from both sides.
C.The function's continuity at x=c is unchanged.
D.The function becomes continuous from the right.
Challenging
The graph of a function f(x) is continuous on the interval (-infinity, 2) and on (2, infinity). At x=2, lim_{x -> 2^-} f(x) = 5 and lim_{x -> 2^+} f(x) = -1. To make f(x) continuous from the left at x=2, how must f(2) be defined?
A.f(2) must be defined as 5.
B.f(2) must be defined as -1.
C.f(2) can be any value, as long as it is defined.
D.It is impossible to make it continuous from the left.
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