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 = 4 c=4 c = 4

Here we want to 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 our function when x is equal to 4 because that's our specified c value. Now looking at our graph here, we can go ahead and just test if this is continuous or not by tracing our function on our graph. Now if I try to trace my function here as I get to x equals 4, I do have to pick up my pen, put it somewhere entirely put it entirely somewhere else, and then continue on. Now because I couldn't just continuously trace with my pen, I know that this function is discontinuous at x equals 4 and I could further verify this numerically by finding that my limit here is actually equal to 1 while my function value is equal to 4. Now these 2 are not the same so that further proves that my function is discontinuous. 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
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Transformations
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Combining Functions
27m

Exponent Rules
32m

Exponential Functions
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Logarithmic Functions
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Properties of Logarithms
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Exponential & Logarithmic Equations
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1. Limits and Continuity2h 2m

Introduction to Limits
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Finding Limits Algebraically
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Continuity
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2. Intro to Derivatives1h 33m

Tangent Lines and Derivatives
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Derivatives as Functions
14m

Basic Graphing of the Derivative
23m

Differentiability
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3. Techniques of Differentiation2h 18m

Basic Rules of Differentiation
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Higher Order Derivatives
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Product and Quotient Rules
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The Chain Rule
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Derivatives of Exponential & Logarithmic Functions
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Related Rates
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Applied Optimization
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Antiderivatives
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Riemann Sums
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22m

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17m

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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 $x=c$ .
$c=4$

Continuous

Discontinuous

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Verified step by step guidance

Step 1: Recall the definition of continuity. A function is continuous at a point 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 = 4. Observe the behavior of the function f(x) near x = 4. Check if there is a defined value for f(4) by looking for a filled dot at x = 4 on the graph.

Step 3: Check the limit of f(x) as x approaches 4 from both the left and the right. Observe the graph to see if the left-hand limit (as x approaches 4 from values less than 4) and the right-hand limit (as x approaches 4 from values greater than 4) are equal.

Step 4: Compare the value of f(4) (if defined) with the limit of f(x) as x approaches 4. If the limit exists and matches the value of f(4), the function is continuous at x = 4. If not, it is discontinuous.

Step 5: Based on the graph, determine if any of the three conditions for continuity are violated at x = 4. If any condition is violated, conclude that the function is discontinuous at x = 4.