Differentiability and Continuity on an Interval

...

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 marked points indicating continuity and differentiability over the interval from -2 to 3.

Accepted Answer

Welcome back, everyone. Consider the function geographed over the interval from negative 2 to 2. Determine the interval where the function G of X is neither continuous nor differentiable. Here we have our graph of GFX, and for our answer choices, it says that our function is neither continuous nor differentiable at the interval or at 0, sorry. B says it's from open bracket negative to 0 closed parentheses union. With open parentheses 01 closed parentheses union with open parentheses 12 closed bracket. C says it's neither continuous nor differentiable at 0 and 1 and D says none. In other words, it's continuous and differentiable throughout the entire interval. Now how can we know whether our function is continuous or differentiable? Well, we know that a function is continuous essentially if it passes the pen test. In other words, if we can run our pen through the function itself where there are no jumps or discontinuities, then we know that it is continuous. And then a function is differentiable if it is continuous and has a well-defined derivative, that is no sharp corners, jumps, or vertical tangents. So let's first try and find out where our function is not continuous and then if it is also not differentiable at those points. No, from our graph here, notice that our function appears to be discontinuous at X equals 0 because there's a hole there, and at X equals 1, because there's also a hole at that point, OK? So we know that geoic is discontinuous. Add x equals 0 and X equals 1. Now the question is, is it differentiable at these points? Well, notice that at X equals 0, our graph appears to not be differentiable because there's a jump discontinuity. Likewise, it's the same thing at X equals 1, OK? So, we also know then. Uh, geopics is not differentiable. At X equals 0 and X equals 1. No, there's also something to note. You might be wondering here. You might be saying, well, there is a sharp corner at X equals 1. So is it not differentiable here? Well yes, it's true. Our function is not differentiable at X E equals -1, OK. And X E equals -1. But no, if we go back to our problem statement, we're looking at where it's neither continuous nor differentiable, and that would be the case where X E equals 0 and X equals 1. It doesn't satisfy either of those conditions at that point. Therefore, this tells us if we go to our answer choices that C is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

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