Determine if the graph of the function f ( x ) f\left(x\right) f ( x ) is continuous and/or differentiable at x = 1 x=1 x = 1 .

this problem, we're asked to determine if the graph of the function f of x is continuous and or differentiable at x equals 1. Now the way that I can do this is by first seeing if it's continuous. And to determine whether or not this is continuous, well, what I can do is I can trace this function with my pen. Now I can see that this is this goes infinitely to the left here, so I just trace it like this from left to right. And then what I notice is when I get to this point, I have to pick up my pen and go down here to keep tracing the function. Because I have to pick up my pen at a certain point and I can't just do this all in one stroke, that means our function is not continuous. And if the function is not continuous, then by default, it's not going to be differentiable. So it is neither differentiable nor continuous, and that would be the solution. Hope you found this video helpful.

2. Intro to Derivatives

Differentiability

Business Calculus

2. Intro to Derivatives

Differentiability

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Multiple Choice

Determine if the graph of the function $f\left(x\right)$ is continuous and/or differentiable at $x=1$ .

Continuous and non-differentiable

Continuous and differentiable

Discontinuous and non-differentiable

Discontinuous and differentiable

Verified step by step guidance

Step 1: Analyze the graph of the function f(x) at x=1. Observe that there is a jump discontinuity at x=1, as the graph has a gap between the left-hand limit and the right-hand limit.

Step 2: Recall the definition of continuity. A function is continuous at a point if the left-hand limit, right-hand limit, and the value of the function at that point are all equal. Since there is a jump discontinuity at x=1, the function is not continuous at this point.

Step 3: Recall the definition of differentiability. A function is differentiable at a point if it is continuous at that point and has a well-defined derivative. Since the function is not continuous at x=1, it cannot be differentiable at this point.

Step 4: Conclude that the function f(x) is discontinuous and non-differentiable at x=1 based on the analysis of the graph and the definitions of continuity and differentiability.

Step 5: Verify the reasoning by observing the graph again. The jump discontinuity confirms the lack of continuity, and the absence of continuity confirms the lack of differentiability at x=1.