Use the graph of f ( x ) f\left(x\right) f ( x ) to determine if the function is continuous or discontinuous at x = c x=c x = c . c = − 2 c=-2 c = − 2

Here we wanna use the graph of f of x to determine if our function is continuous or discontinuous at x equals c. Now here we're looking at x equals negative 2 because that's our specified value of c. Now looking at my graph here, I know that if I trace along my function at x equals 2, I don't have to lift my pen at all. So I know that this function is continuous. Now, of course, we can also verify this by looking at both the limit and the function value at this point and we would see that they are equal to each other because they are both 2. Thanks for watching, and let me know if you have questions.

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0. Functions4h 53m

Introduction to Functions
16m

Piecewise Functions
10m

Properties of Functions
9m

Common Functions
1h 8m

Transformations
5m

Combining Functions
27m

Exponent Rules
32m

Exponential Functions
28m

Logarithmic Functions
24m

Properties of Logarithms
34m

Exponential & Logarithmic Equations
35m

1. Limits and Continuity2h 2m

Introduction to Limits
41m

Finding Limits Algebraically
40m

Continuity
40m

2. Intro to Derivatives1h 33m

Tangent Lines and Derivatives
30m

Derivatives as Functions
14m

Basic Graphing of the Derivative
23m

Differentiability
26m

3. Techniques of Differentiation2h 18m

Basic Rules of Differentiation
43m

Higher Order Derivatives
12m

Product and Quotient Rules
55m

The Chain Rule
27m

4. Derivatives of Exponential & Logarithmic Functions1h 16m

Derivatives of Exponential & Logarithmic Functions
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5. Applications of Derivatives2h 19m

Implicit Differentiation
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Logarithmic Differentiation
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Related Rates
59m

Differentials
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Linearization
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6. Graphical Applications of Derivatives6h 0m

Intro to Extrema
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The First Derivative Test
1h 20m

Concavity
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The Second Derivative Test
22m

Curve Sketching
44m

Finding Global Extrema
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Applied Optimization
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Antiderivatives
17m

Indefinite Integrals
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8. Definite Integrals4h 36m

Estimating Area with Finite Sums
42m

Riemann Sums
1h 16m

Introduction to Definite Integrals
22m

Fundamental Theorem of Calculus
30m

Average Value of a Function
17m

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55m

Substitution for Definite Integrals
32m

9. Graphical Applications of Integrals1h 43m

Area Between Curves
1h 19m

Introduction to Volume & Disk Method
23m

10. Integrals of Inverse, Exponential, & Logarithmic Functions21m

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11. Techniques of Integration2h 7m

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Angles
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46m

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16. Probability & Calculus45m

Continuous Probability Models
45m

1. Limits and Continuity

Continuity

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Continuity

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Multiple Choice

Use the graph of $f$\left$(x$\right$)$ to determine if the function is continuous or discontinuous at $x=c$ .
$c=-2$

Continuous

Discontinuous

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Verified step by step guidance

Step 1: Recall the definition of continuity at a point x = c. A function f(x) is continuous at x = c if the following three conditions are satisfied: (1) f(c) is defined, (2) the limit of f(x) as x approaches c exists, and (3) the limit of f(x) as x approaches c is equal to f(c).

Step 2: Analyze the graph at x = -2. Check if there is a point on the graph at x = -2. From the graph, observe that there is a filled circle at x = -2, indicating that f(-2) is defined.

Step 3: Determine if the limit of f(x) as x approaches -2 exists. Observe the behavior of the graph as x approaches -2 from both the left and the right. The graph approaches the same y-value from both sides, indicating that the limit exists.

Step 4: Verify if the limit of f(x) as x approaches -2 is equal to f(-2). From the graph, the y-value of the function at x = -2 matches the value approached by the graph from both sides, satisfying the third condition for continuity.

Step 5: Conclude that the function f(x) is continuous at x = -2 because all three conditions for continuity are satisfied.