Science:Math Exam Resources/Courses/MATH110/December 2010/Question 01 (b)

MATH110 December 2010

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Question 01 (b)
Determine whether each statement is true. If it is, explain why; if not, give a counter-example. Not all continuous functions are differentiable.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1

If you think the statement is true, that is not all continuous function are differentiable, you would need to give an example showing a continuous function that is not differentiable. If you believe the statement is false, that is all continuous functions are differentiable, you would need to show that given any continuous function, then you must have that it is differentiable [everywhere]. As always, it is good to try a few examples to get a feeling for this problem first before trying to answer it.

Hint 2
Think about what makes a function non-differentiable. Is there a function that has this property but is still continuous?

Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. The most common reasons that a function $f$ is non-differentiable at a point x = a are as follows: a) $f$ has a sharp corner (called a cusp) at x = a, where the tangent lines from both sides of a converge to different values. b) $f$ has a vertical tangent line at x = a, that is, the slope of the tangent line at a is infinity. Both of these conditions can occur for continuous functions. For example: a) The function $f(x)=|x|$ is continuous, but has a sharp corner at x = 0. b) The function $f(x)={\sqrt[{3}]{x}}$ is continuous, but has a vertical tangent line at x = 0. Thus it is clear that there are continuous functions that are not differentiable, so the statement Not all continuous functions are differentiable is true. To prove this, it is sufficient to give an example of a function that is continuous but not differentiable, like one of the functions shown above.

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