How do you know if a function is differentiable on a closed interval?

A function is “differentiable” over an interval if that function is both continuous, and has only one output for every input. Another way of saying this is for every x input into the function, there is only one value of y (i.e. no vertical lines, function overlapping itself, etc).

Can a function be continuous and differentiable on a closed interval?

The Mean Value Theorem states that if a function f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists a point c in the interval (a,b) such that f'(c) is equal to the function’s average rate of change over [a,b].

What makes a function differentiable on an open interval?

The definition of differentiability is expressed as follows: f is differentiable on an open interval (a,b) if lim h → 0 f ( c + h ) − f ( c ) h exists for every c in (a,b).

Does the mean value theorem have to be on a closed interval?

A and B are given in a closed interval in the question because the hypotheses of the mean value theorem include that the function must be continuous on the interval [A,B] and differentiable on the interval (A,B).

How do you show that a function is continuous on an open interval?

A function is said to be continuous on an interval when the function is defined at every point on that interval and undergoes no interruptions, jumps, or breaks. If some function f(x) satisfies these criteria from x=a to x=b, for example, we say that f(x) is continuous on the interval [a, b].

What makes a function continuous but not differentiable?

The absolute value function is continuous (i.e. it has no gaps). It is differentiable everywhere except at the point x = 0, where it makes a sharp turn as it crosses the y-axis. A cusp on the graph of a continuous function. At zero, the function is continuous but not differentiable.

Does a function have to be continuous to be differentiable?

We see that if a function is differentiable at a point, then it must be continuous at that point. If is not continuous at , then is not differentiable at . Thus from the theorem above, we see that all differentiable functions on are continuous on .

How do I know if a function is differentiable?

A function is said to be differentiable if the derivative of the function exists at all points in its domain. Particularly, if a function f(x) is differentiable at x = a, then f′(a) exists in the domain.

Can a function be differentiable but not continuous?

Why do we define differentiability on open intervals and not closed intervals?

The reason that so many theorems require a function to be continuous on [a,b] and differentiable on (a,b) is not that differentiability on [a,b] is undefined or problematic; it is that they do not need differentiability in any sense at the endpoints, and by using this looser phrasing the theorem becomes more generally applicable.

Can a function be differentiable at the end?

$\\begingroup$ Depends exactly on what you mean. At first differentiability is only defined for interior points. However, one is then typically introduced to the concept of lateral derivative and with that the definition of differentiability is extended to the end points.

Can a derivative be defined at the end of an interval?

So the answer is yes: You can define the derivative in a way, such that f ′ is also defined for the end points of a closed interval. Note that for some theorem like the mean value theorem you only need continuity at the end points of the interval. Thanks for contributing an answer to Mathematics Stack Exchange!

Is there a more general definition of differentiability?

It depends on the definition of differentiability you have. Some textbooks only define it for interior points. But there is also a more general definition (see this answer for references):

How do you know if a function is differentiable on a closed interval? A function is “differentiable” over an interval if that function is both continuous, and has only one output for every input. Another way of saying this is for every x input into the function, there is only one value of y (i.e.…