Quick Answer: How Do You Know If A Function Is Differentiable On An Interval?

What does it mean for a function to be differentiable on an interval?

The problem with this approach, though, is that some functions have one or many points or intervals where their derivatives are undefined.

A function f is differentiable at a point c if.


Similarly, f is differentiable on an open interval (a, b) if.

exists for every c in (a, b)..

Can a function be differentiable on a closed 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.

How do you know if a function is continuous on an 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].

Does a function have to be continuous to be differentiable?

In particular, any differentiable function must be continuous at every point in its domain. The converse does not hold: a continuous function need not be differentiable. For example, a function with a bend, cusp, or vertical tangent may be continuous, but fails to be differentiable at the location of the anomaly.

How do you know if a function is not differentiable?

We can say that f is not differentiable for any value of x where a tangent cannot ‘exist’ or the tangent exists but is vertical (vertical line has undefined slope, hence undefined derivative).

How can a function fail to be differentiable?

Three Basic Ways a Function Can Fail to be DifferentiableThe function may be discontinuous at a point.The function may have a corner (or cusp) at a point.The function may have a vertical tangent at a point.

How do you show that a function is differentiable continuous?

Page 1Differentiable Implies Continuous. Theorem: If f is differentiable at x0, then f is continuous at x0. … number – this won’t change its value. lim f(x) – f(x0) = lim. … = f (x) 0· = 0. (Notice that we used our assumption that f was differentiable when we wrote down f (x).)

How do you know if a function is differentiable?

Lesson 2.6: Differentiability: A function is differentiable at a point if it has a derivative there. … Example 1: … If f(x) is differentiable at x = a, then f(x) is also continuous at x = a. … f(x) − f(a) … (f(x) − f(a)) = lim. … (x − a) · f(x) − f(a) x − a This is okay because x − a �= 0 for limit at a. … (x − a) lim. … f(x) − f(a)More items…

Can a function be differentiable and not continuous?

When a function is differentiable it is also continuous. But a function can be continuous but not differentiable. For example the absolute value function is actually continuous (though not differentiable) at x=0.

Are functions differentiable at endpoints?

On the real line, a function is differentiable if and only if it is both left and right differentiable, and those two derivatives agree. At the left endpoint, the left derivative doesn’t exist. At the right endpoint, the right derivative doesn’t exist.

Are all polynomial functions differentiable?

All of the standard functions are differentiable except at certain singular points, as follows: Polynomials are differentiable for all arguments. A rational function is differentiable except where q(x) = 0, where the function grows to infinity.

Why are functions not differentiable at corners?

A function is not differentiable at a if its graph has a corner or kink at a. … Since the function does not approach the same tangent line at the corner from the left- and right-hand sides, the function is not differentiable at that point.

What makes a function not differentiable?

A function which jumps is not differentiable at the jump nor is one which has a cusp, like |x| has at x = 0. Generally the most common forms of non-differentiable behavior involve a function going to infinity at x, or having a jump or cusp at x.

What does it mean when a graph is differentiable?

A function is differentiable at a point when there’s a defined derivative at that point. This means that the slope of the tangent line of the points from the left is approaching the same value as the slope of the tangent of the points from the right.

How do you know if a function is continuous?

If a function f is only defined over a closed interval [c,d] then we say the function is continuous at c if limit(x->c+, f(x)) = f(c). Similarly, we say the function f is continuous at d if limit(x->d-, f(x))= f(d).