Differentiability and Continuity on an Interval

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Question

Differentiability and Continuity on an Interval

Each figure in Exercises 45–50 shows the graph of a function over a closed interval D. At what domain points does the function appear to be

c. neither continuous nor differentiable?

Give reasons for your answers.

Graph of a function with open circles at x = -3 and x = 2, indicating points of discontinuity on the interval from -3 to 3.

Accepted Answer

Below there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Consider the function GFX graphed over the interval square bracket 4.4 square bracket. Determine the interval where the function G of X is neither continuous nor differentiable. Awesome. So it appears for this particular problem we're ultimately trying to determine the interval where the function G of X is neither continuous nor differentiable, and this is our final answer that we're ultimately trying to solve for. So with that said, let's quickly read off our multiple choice answers to see what our final answer might be. A is square bracket 4.4 square bracket. B is curly bracket 0 curly bracket. C is square bracket 4.0 04 square and D is none. Awesome. So now, first off, in order to solve this problem, let's take a moment to investigate our graph of Y equals GFX to help us better visualize what's happening in this particular problem. And as you can see, we have our curve represented in red, and we have what appears to be two solid points and two holes, and both holes are located on the Y axis at what appears to be. 03, and what appears to be at 03 and 0, so once again at 0, actually not 3, it looks like 2.5, means it's halfway between 2 and 3, and at at 0, negative 2.5. And then we have two solid points that touch the X axis at. At what appears to be 3.5.0 and 3.5.0. Awesome. So we need to note and recall that a function is differentiable, as we should recall, if it is continuous and has a well-defined derivative, meaning there are no sharp corners. jumps or vertical tangents. And however, as we can see from our graph, the function appears to be discontinuous, it's discontinuous at X equals 0 because there is a hole at X equals 0. So there's another hole, so I said there are 2 holes, but there's technically 3, because we have one at the origin point as well. And because there's a discontinuity or this hole at X equals 0. And because there's this hole, we need to note that from the graph, the function appears to be not differentiable at this value of X equals 0, because once again it's a hole, and because there's a jump discontinuity at X equals 0. So that will mean That without a doubt, our final answer has to be multiple choice answer letter B, which once again is curly bracket 0, curly bracket, and that's it. That's our final answer. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Differentiability and Continuity on an Interval

Each figure in Exercises 45–50 shows the graph of a function over a closed interval D. At what domain points does the function appear to be

c. neither continuous nor differentiable?

Give reasons for your answers.

Key Concepts

Continuity