Continuous function which is not differentiable

Cancel OK. Differentiability Implies Continuity If is a differentiable function at , then is continuous at. To explain why this is true, we are going to use the following definition of the derivative Assuming that exists, we want to show that is continuous at , hence we must show that Starting with we multiply and divide by to get Since we apply the Difference Law to the left hand side and use continuity of a constant to obtain that Next, we add on both sides and get that Now we see that , and so is continuous at.

If is not continuous at , then is not differentiable at. Which of the following functions are continuous but not differentiable on? Consider What values of and make differentiable at? To start, we know that must be continuous at , since it has to be differentiable there. We will start by making continuous at. Write with me: So for the function to be continuous, we must have We also must ensure that the function is differentiable at. In other words, we have to ensure that the following limit exists In order to compute this limit, we have to compute the two one-sided limits and since changes expression at.

Write with me and Hence, we must have Ah! So now the equation that must be satisfied Therefore,. We can easily observe that the absolute value graph is continuous as we can draw the graph without picking up your pencil. But we can also quickly see that the slope of the curve is different on the left as it is on the right.

This suggests that the instantaneous rate of change is different at the vertex i. We use one-sided limits and our definition of derivative to determine whether or not the slope on the left and right sides are equal. While the function is continuous, it is not differentiable because the derivative is not continuous everywhere, as seen in the graphs below.

Get access to all the courses and over HD videos with your subscription. Get My Subscription Now. I think Wikipedia calls it the "Blancmange curve". I like this one better than the Weierstrass function, but this is personal preference.

I really like this answer I went over in my lecture:. Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. Are there any functions that are always continuous yet not differentiable?

Or vice-versa? Ask Question. Asked 11 years, 3 months ago. Active 5 months ago. Viewed 11k times. Justin L. The Weierstrass function has been very nicely identified in the answers below, and it is an important counter-example that comes up immediately in advanced calculus. Add a comment.