What does differentiable and continuous mean

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We see that if a function is differentiable at a point, then it must be continuous at that point. 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.

Thus from the theorem above, we see that all differentiable functions on are continuous on. Nevertheless there are continuous functions on that are not differentiable on. So for the function to be continuous, we must have We also must ensure that the function is differentiable at. No, those are not all equivalent. The first two are; the third is weaker. So "continuously differentiable" means "differentiable in a continuous way". Add a comment. Active Oldest Votes.

Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name. Email Required, but never shown. 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. Please click here if you are not redirected within a few seconds. Home » Derivatives » Continuity and Differentiability. But the reverse need not hold. Not that if that fraction does converge, the numerator necessarily converges to zero, implying continuity as I mentioned in the first paragraph.

Continuity means that the limit from both sides of a value is equal to the function's value at that point. This is stronger than not continuous, which in turn is stronger than not differentiable.

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. What is the difference between "differentiable" and "continuous" Ask Question. Asked 7 years, 7 months ago.