Continuity Conditions

Why This Matters

Continuity is the foundation that makes calculus work. When you're tested on limits, derivatives, and integrals, you're constantly being asked to verify whether certain theorems apply—and almost every major theorem in calculus (the Mean Value Theorem, the Extreme Value Theorem, the Intermediate Value Theorem) requires continuity as a precondition. If you can't quickly assess whether a function is continuous, you'll struggle to justify your answers on FRQs and waste precious time on multiple choice.

The AP exam doesn't just want you to recite the three-part definition of continuity. You're being tested on your ability to classify discontinuities, verify theorem conditions, and connect continuity to differentiability. Think of continuity as your gateway skill: it determines which calculus tools you're allowed to use. Don't just memorize the conditions—know what breaks continuity, what type of break it is, and what that means for the theorems you want to apply.

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The Three-Part Definition

Every continuity question on the AP exam traces back to one fundamental idea: a function is continuous at a point if you can draw through it without lifting your pencil, AND the function value matches where the limit is heading.

Continuity at a Point

• Three conditions must ALL hold at $x = c$: the function $f(c)$ must be defined, $\lim_{x \to c} f(x)$ must exist, and these two values must be equal
• The limit existing means both one-sided limits agree—if they don't match, the overall limit doesn't exist
• Written compactly: $\lim_{x \to c} f(x) = f(c)$, which the AP exam often asks you to verify step-by-step

One-Sided Continuity

• Left-hand continuity requires $\lim_{x \to c^-} f(x) = f(c)$—the function approaches from below and lands on the point
• Right-hand continuity requires $\lim_{x \to c^+} f(x) = f(c)$—the function approaches from above and lands on the point
• Full continuity at $c$ demands both one-sided limits exist, equal each other, AND equal $f(c)$

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Types of Discontinuities

When continuity fails, it fails in predictable ways. The AP exam expects you to classify discontinuities, not just identify them. Each type tells you exactly which part of the three-part definition broke down.

Removable Discontinuity

• The limit exists but doesn't equal $f(c)$—either $f(c)$ is undefined or it's defined as the "wrong" value
• Graphically appears as a hole in the curve, sometimes with a separate point plotted elsewhere
• Called "removable" because you could redefine $f(c)$ to equal the limit and "fix" the function

Jump Discontinuity

• Left and right limits both exist but aren't equal—the function literally "jumps" from one value to another
• The overall limit $\lim_{x \to c} f(x)$ does not exist because the two-sided limit requires agreement
• Common in piecewise functions where different formulas apply on each side of the boundary

Infinite Discontinuity

• The function approaches $\pm\infty$ as $x \to c$, creating a vertical asymptote
• The limit does not exist in the traditional sense—though we say the limit "is infinite" to describe the behavior
• Typical culprit: rational functions where the denominator equals zero but the numerator doesn't

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Continuity on Intervals and Domains

Individual point continuity scales up to intervals and entire domains. This is where the big theorems live—and where the exam tests whether you understand their hypotheses.

Continuity on an Interval

• Continuous on an open interval $(a, b)$ means continuous at every point inside—no endpoint conditions needed
• Continuous on a closed interval $[a, b]$ adds requirements: right-continuous at $a$, left-continuous at $b$
• Closed-interval continuity unlocks the Extreme Value Theorem and is required for the Mean Value Theorem

Elementary Function Continuity

• Polynomials are continuous everywhere—no restrictions, no discontinuities, always safe to apply theorems
• Rational, trig, exponential, and log functions are continuous on their natural domains—watch for where they're undefined
• Knowing these defaults lets you skip verification steps and move faster on the exam

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Building Continuous Functions

Continuous functions combine nicely. This matters because complex functions are built from simpler pieces, and you need to know when continuity is preserved.

Algebraic Combinations

• Sums, differences, products, and quotients of continuous functions remain continuous—except division by zero
• If $f$ and $g$ are continuous at $c$, then $f + g$, $f - g$, $fg$, and $\frac{f}{g}$ (when $g(c) \neq 0$) are continuous at $c$
• This justifies treating complicated expressions as continuous without checking each piece separately

Composite Functions

• If $g$ is continuous at $c$ and $f$ is continuous at $g(c)$, then $f(g(x))$ is continuous at $c$
• The "chain" of continuity must be unbroken—continuity of the outer function must hold at the output of the inner function
• Critical for functions like $\sin(x^2)$, $e^{\cos x}$, or $\ln(x + 1)$—verify the inner function's output stays in the outer function's domain

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Theorems That Require Continuity

These theorems appear constantly on the AP exam. Each one has continuity as a non-negotiable hypothesis—if continuity fails, the theorem doesn't apply.

Intermediate Value Theorem (IVT)

• States that if $f$ is continuous on $[a, b]$ and $N$ is between $f(a)$ and $f(b)$, then $f(c) = N$ for some $c \in (a, b)$
• Guarantees existence of solutions—the function must hit every $y$-value between its endpoints
• Classic FRQ setup: show a continuous function changes sign on an interval to prove a root exists

Extreme Value Theorem (EVT)

• States that if $f$ is continuous on a closed interval $[a, b]$, then $f$ attains both a maximum and minimum value
• Justifies the Candidates Test—you're guaranteed the absolute extrema exist, so checking critical points and endpoints will find them
• Both conditions matter: continuous AND closed interval—open intervals or discontinuous functions can have no max or min

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Continuity and Differentiability

This relationship is tested constantly. Understand the one-way implication and the counterexamples.

The Implication

• Differentiability implies continuity—if $f'(c)$ exists, then $f$ must be continuous at $c$
• Continuity does NOT imply differentiability—a function can be continuous but have no derivative at certain points
• The contrapositive is useful: if $f$ is discontinuous at $c$, then $f$ is definitely not differentiable at $c$

Continuous but Not Differentiable

• Sharp corners (like $f(x) = |x|$ at $x = 0$) are continuous but have different left and right derivatives
• Cusps and vertical tangents represent points where the derivative becomes infinite or undefined
• These points are still valid critical points for optimization—$f'(c)$ undefined counts as a critical number

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Quick Reference Table

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Self-Check Questions

1. A piecewise function has $\lim_{x \to 3^-} f(x) = 5$ and $\lim_{x \to 3^+} f(x) = 5$, but $f(3) = 7$. What type of discontinuity is this, and which part of the three-part definition fails?

2. Compare the hypotheses of the Mean Value Theorem and the Extreme Value Theorem. Which theorem has the stricter requirements, and why?

3. If you know that $g(x)$ is differentiable at $x = 2$, what can you conclude about the continuity of $g$ at that point? What about the converse?

4. A student claims that since $f(x) = \frac{x^2 - 4}{x - 2}$ is undefined at $x = 2$, the function has an infinite discontinuity there. Identify the error and classify the discontinuity correctly.

5. For the function $f(x) = |x - 1|$, explain why the Extreme Value Theorem applies on $[-2, 3]$ but the Mean Value Theorem does not. What specific condition fails for MVT?