Quick Answer: What Are The Conditions For A Function To Be Differentiable?

Is a function differentiable at a hole?


A function with a removable discontinuity at the point is not differentiable at since it’s not continuous at .

Thus, is not differentiable.

However, you can take an arbitrary differentiable function ..

Do limits exist at corners?

what is the limit. The limit is what value the function approaches when x (independent variable) approaches a point. takes only positive values and approaches 0 (approaches from the right), we see that f(x) also approaches 0. … exist at corner points.

What two conditions must be met for a function to be differentiable?

In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its domain. A differentiable function is smooth (the function is locally well approximated as a linear function at each interior point) and does not contain any break, angle, or cusp.

How do you differentiate a function?

Apply the power rule to differentiate a function. The power rule states that if f(x) = x^n or x raised to the power n, then f'(x) = nx^(n – 1) or x raised to the power (n – 1) and multiplied by n. For example, if f(x) = 5x, then f'(x) = 5x^(1 – 1) = 5.

Is continuity necessary for differentiability?

The converse to the Theorem is false. A continuous function need not be differentiable. In other words, differentiability is a stronger condition than continuity.

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…

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.

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.

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).

What does it mean for a function to be 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 tell if a function is not differentiable on a graph?

If there derivative can’t be found, or if it’s undefined, then the function isn’t differentiable there. So, for example, if the function has an infinitely steep slope at a particular point, and therefore a vertical tangent line there, then the derivative at that point is undefined.

Why does a function have to be continuous to be differentiable?

Until then, intuitively, a function is continuous if its graph has no breaks, and differentiable if its graph has no corners and no breaks. So differentiability is stronger. A function is only differentiable on an open set, then it has no sense to say that your function is differentiable en a or on b.

Can a function be not continuous?

If they are equal the function is continuous at that point and if they aren’t equal the function isn’t continuous at that point. … The function value and the limit aren’t the same and so the function is not continuous at this point. This kind of discontinuity in a graph is called a jump discontinuity.

What is differentiability and continuity?

Continuity of a function is the characteristic of a function by virtue of which, the graphical form of that function is a continuous wave. A differentiable function is a function whose derivative exists at each point in its domain.