Defining Continuity at a Point - AP Calculus AB
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State the definition of a removable discontinuity.
State the definition of a removable discontinuity.
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A point where $f(x)$ is not defined or $\text{lim}_{x \to c} f(x) \neq f(c)$, but can be redefined. The gap can be filled by redefining the function at that point.
A point where $f(x)$ is not defined or $\text{lim}_{x \to c} f(x) \neq f(c)$, but can be redefined. The gap can be filled by redefining the function at that point.
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Verify continuity of $f(x) = \frac{1}{x^2}$ at $x = 0$.
Verify continuity of $f(x) = \frac{1}{x^2}$ at $x = 0$.
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$f(x)$ is not continuous at $x = 0$. Function undefined at $x = 0$ (division by zero).
$f(x)$ is not continuous at $x = 0$. Function undefined at $x = 0$ (division by zero).
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Find if $f(x) = \frac{x - 1}{x^2 - 1}$ is continuous at $x = 1$.
Find if $f(x) = \frac{x - 1}{x^2 - 1}$ is continuous at $x = 1$.
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Removable discontinuity at $x = 1$. Factor gives $\text{lim}_{x \to 1} \frac{1}{x+1} = \frac{1}{2}$ but $f(1)$ undefined.
Removable discontinuity at $x = 1$. Factor gives $\text{lim}_{x \to 1} \frac{1}{x+1} = \frac{1}{2}$ but $f(1)$ undefined.
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What is the third condition for continuity at a point $x = c$?
What is the third condition for continuity at a point $x = c$?
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$\lim_{x \to c} f(x) = f(c)$. The limit value must equal the function value.
$\lim_{x \to c} f(x) = f(c)$. The limit value must equal the function value.
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What is a continuous function?
What is a continuous function?
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A function that is continuous at every point in its domain. No breaks, holes, or jumps anywhere in the domain.
A function that is continuous at every point in its domain. No breaks, holes, or jumps anywhere in the domain.
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Determine if $f(x) = \text{sin}(x)$ is continuous for all $x$.
Determine if $f(x) = \text{sin}(x)$ is continuous for all $x$.
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$f(x)$ is continuous for all $x$. Trigonometric functions are continuous on their domains.
$f(x)$ is continuous for all $x$. Trigonometric functions are continuous on their domains.
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What is the conclusion if $\text{lim}_{x \to c} f(x) = f(c)$ and $f(c)$ is defined?
What is the conclusion if $\text{lim}_{x \to c} f(x) = f(c)$ and $f(c)$ is defined?
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The function $f(x)$ is continuous at $x = c$. All three conditions for continuity are satisfied.
The function $f(x)$ is continuous at $x = c$. All three conditions for continuity are satisfied.
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Which condition fails if $\text{lim}_{x \to c} f(x)$ does not exist?
Which condition fails if $\text{lim}_{x \to c} f(x)$ does not exist?
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The condition that $\text{lim}_{x \to c} f(x)$ exists. The left and right limits don't agree.
The condition that $\text{lim}_{x \to c} f(x)$ exists. The left and right limits don't agree.
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What is the first condition for continuity at a point $x = c$?
What is the first condition for continuity at a point $x = c$?
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$f(c)$ is defined. The function must have a value at point $c$.
$f(c)$ is defined. The function must have a value at point $c$.
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Determine if $f(x) = \frac{1}{x}$ is continuous for $x \neq 0$.
Determine if $f(x) = \frac{1}{x}$ is continuous for $x \neq 0$.
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$f(x)$ is continuous for $x \neq 0$. Rational functions are continuous except where denominator is zero.
$f(x)$ is continuous for $x \neq 0$. Rational functions are continuous except where denominator is zero.
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State the definition of a jump discontinuity.
State the definition of a jump discontinuity.
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A point where left and right limits exist but are not equal. The function jumps from one value to another at that point.
A point where left and right limits exist but are not equal. The function jumps from one value to another at that point.
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State the Intermediate Value Theorem (IVT) condition for continuity.
State the Intermediate Value Theorem (IVT) condition for continuity.
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If $f$ is continuous on $[a, b]$ and $N$ is between $f(a)$ and $f(b)$, then $\text{exists } c \text{ in } (a, b)$ with $f(c) = N$. Continuous functions take on all intermediate values.
If $f$ is continuous on $[a, b]$ and $N$ is between $f(a)$ and $f(b)$, then $\text{exists } c \text{ in } (a, b)$ with $f(c) = N$. Continuous functions take on all intermediate values.
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State whether $f(x) = \frac{\text{sin}(x)}{x}$ is continuous at $x = 0$.
State whether $f(x) = \frac{\text{sin}(x)}{x}$ is continuous at $x = 0$.
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Removable discontinuity at $x = 0$. $\text{lim}_{x \to 0} \frac{\text{sin}(x)}{x} = 1$ but $f(0)$ undefined.
Removable discontinuity at $x = 0$. $\text{lim}_{x \to 0} \frac{\text{sin}(x)}{x} = 1$ but $f(0)$ undefined.
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Evaluate continuity of piecewise function: $f(x) = x^2$ for $x \neq 1$, $f(1) = 1$.
Evaluate continuity of piecewise function: $f(x) = x^2$ for $x \neq 1$, $f(1) = 1$.
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Continuous at $x = 1$. $\text{lim}_{x \to 1} x^2 = 1$ equals $f(1) = 1$.
Continuous at $x = 1$. $\text{lim}_{x \to 1} x^2 = 1$ equals $f(1) = 1$.
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Check continuity of $f(x) = \frac{1}{x}$ at $x = 1$.
Check continuity of $f(x) = \frac{1}{x}$ at $x = 1$.
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Continuous at $x = 1$. All conditions satisfied: defined, limit exists, and they're equal.
Continuous at $x = 1$. All conditions satisfied: defined, limit exists, and they're equal.
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Given $f(x) = x^2$ for $x \neq 1$ and $f(1) = 3$, determine continuity at $x = 1$.
Given $f(x) = x^2$ for $x \neq 1$ and $f(1) = 3$, determine continuity at $x = 1$.
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Not continuous at $x = 1$. Limit is $1$ but function value is $3$.
Not continuous at $x = 1$. Limit is $1$ but function value is $3$.
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Given $f(x) = |x|$, determine continuity at $x = 0$.
Given $f(x) = |x|$, determine continuity at $x = 0$.
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$f(x)$ is continuous at $x = 0$. Left limit equals right limit equals $f(0) = 0$.
$f(x)$ is continuous at $x = 0$. Left limit equals right limit equals $f(0) = 0$.
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Find if $f(x) = \frac{x^2 - 4}{x - 2}$ is continuous at $x = 2$.
Find if $f(x) = \frac{x^2 - 4}{x - 2}$ is continuous at $x = 2$.
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Removable discontinuity at $x = 2$. Factor gives $\lim_{x \to 2} (x+2) = 4$ but $f(2)$ undefined.
Removable discontinuity at $x = 2$. Factor gives $\lim_{x \to 2} (x+2) = 4$ but $f(2)$ undefined.
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State the definition of an infinite discontinuity.
State the definition of an infinite discontinuity.
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A point where $f(x)$ approaches $\text{+}\text{ or }\text{-}\text{ infinity}$ as $x \to c$. The function grows without bound near that point.
A point where $f(x)$ approaches $\text{+}\text{ or }\text{-}\text{ infinity}$ as $x \to c$. The function grows without bound near that point.
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Which condition fails if $f(x)$ is not defined at $x = c$?
Which condition fails if $f(x)$ is not defined at $x = c$?
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The condition that $f(c)$ is defined. The function has no value at that point.
The condition that $f(c)$ is defined. The function has no value at that point.
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Find the type of discontinuity for $f(x) = \frac{1}{x-1}$ at $x = 1$.
Find the type of discontinuity for $f(x) = \frac{1}{x-1}$ at $x = 1$.
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Infinite discontinuity at $x = 1$. Vertical asymptote creates infinite discontinuity.
Infinite discontinuity at $x = 1$. Vertical asymptote creates infinite discontinuity.
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Determine if $f(x) = x^2 + 2x + 1$ is continuous for all $x$.
Determine if $f(x) = x^2 + 2x + 1$ is continuous for all $x$.
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$f(x)$ is continuous for all $x$. Polynomial functions are continuous everywhere.
$f(x)$ is continuous for all $x$. Polynomial functions are continuous everywhere.
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Determine continuity of $f(x) = \text{e}^x$ for all $x$.
Determine continuity of $f(x) = \text{e}^x$ for all $x$.
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$f(x)$ is continuous for all $x$. Exponential functions are continuous everywhere.
$f(x)$ is continuous for all $x$. Exponential functions are continuous everywhere.
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Identify the limit that must exist for continuity at $x = c$.
Identify the limit that must exist for continuity at $x = c$.
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$\lim_{x \to c} f(x)$ must exist. Both one-sided limits must exist and be equal.
$\lim_{x \to c} f(x)$ must exist. Both one-sided limits must exist and be equal.
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Identify the type of discontinuity: $f(x)$ undefined but limit exists.
Identify the type of discontinuity: $f(x)$ undefined but limit exists.
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Removable discontinuity. Can be fixed by defining the function at that point.
Removable discontinuity. Can be fixed by defining the function at that point.
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True or False: A function can be continuous at a point and not differentiable.
True or False: A function can be continuous at a point and not differentiable.
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True. Functions can be continuous but have corners (like $|x|$ at $x=0$).
True. Functions can be continuous but have corners (like $|x|$ at $x=0$).
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Determine continuity of $f(x) = \text{tan}(x)$ for $x \neq \frac{\text{n}\text{π}}{\text{2}}$.
Determine continuity of $f(x) = \text{tan}(x)$ for $x \neq \frac{\text{n}\text{π}}{\text{2}}$.
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$f(x)$ is continuous for $x \neq \frac{\text{n}\text{π}}{\text{2}}$. Tangent has vertical asymptotes at odd multiples of $\frac{\pi}{2}$.
$f(x)$ is continuous for $x \neq \frac{\text{n}\text{π}}{\text{2}}$. Tangent has vertical asymptotes at odd multiples of $\frac{\pi}{2}$.
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What is the definition of continuity at a point $x = c$?
What is the definition of continuity at a point $x = c$?
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A function $f(x)$ is continuous at $x = c$ if $\text{lim}_{x \to c} f(x) = f(c)$. This states that the limit equals the function value at that point.
A function $f(x)$ is continuous at $x = c$ if $\text{lim}_{x \to c} f(x) = f(c)$. This states that the limit equals the function value at that point.
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Identify the type of discontinuity if $\text{lim}_{x \to c} f(x) = \text{+}\text{ or }\text{-} \text{ infinity}$.
Identify the type of discontinuity if $\text{lim}_{x \to c} f(x) = \text{+}\text{ or }\text{-} \text{ infinity}$.
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Infinite discontinuity. The function approaches infinity at that point.
Infinite discontinuity. The function approaches infinity at that point.
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Is the function $f(x) = [x]$ (greatest integer function) continuous for integer $x$?
Is the function $f(x) = [x]$ (greatest integer function) continuous for integer $x$?
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No, it is not continuous at integer $x$. Floor function has jump discontinuities at every integer.
No, it is not continuous at integer $x$. Floor function has jump discontinuities at every integer.
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