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.

1. Limits and Continuity

Continuity

Business Calculus

1. Limits and Continuity

Continuity

Struggling with Business Calculus?

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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

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.