## Can a function be differentiable but its derivative not continuous?

The basic example of a differentiable function with discontinuous derivative is f(x)={x2sin(1/x)if x≠00if x=0. The differentiation rules show that this function is differentiable away from the origin and the difference quotient can be used to show that it is differentiable at the origin with value f′(0)=0.

### Does differentiable mean the derivative is continuous?

A function is said to be continuously differentiable if its derivative is also a continuous function; there exists a function that is differentiable but not continuously differentiable as being shown below (in the section Differentiability classes).

#### Does a differentiable function have to be continuous?

If a function is differentiable then it’s also continuous. This property is very useful when working with functions, because if we know that a function is differentiable, we immediately know that it’s also continuous.

**Is the derivative necessarily continuous?**

A continuous function is not necessarily differentiable, but a differentiable function is necessarily continuous (at every point where it is differentiable) as being shown below (in the section Differentiability and continuity).

**Are all derivatives continuous?**

The conclusion is that derivatives need not, in general, be continuous! 1 if x > 0. A first impression may bring to mind the absolute value function, which has slopes of −1 at points to the left of zero and slopes of 1 to the right. However, the absolute value function is not differentiable at zero.

## Why are not all continuous functions have derivatives?

No. Since a function has to be both continuous and smooth in order to have a derivative, not all continuous functions are differentiable.

### Is every differentiable function is continuous?

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.

#### When a function is continuous is it differentiable?

As long as a derivate can be found of a function at a certain point, the function is continuous at that point due to the proof aforementioned. If a Function is Continuous, is it Differentiable? A continuous function can be non-differentiable. Any differentiable function is always continuous.

**How do you prove that a derivative is not continuous at 0?**

Then use the definition of the derivative to find f ′ ( 0). You should get f ′ ( 0) = 0 . Then show that f ′ ( x) has no limit as x → 0, so f ′ is not continuous at 0. (Hint: the first term in ( ∗) tends to 0; what happens to the second?)

**Does derivative of differentiable function satisfy intermediate value property?**

The basic idea for the example comes from the observation that derivative of differentiable function satisfies intermediate value property. This means that for any counterexample, the derivative would still satisfy the intermediate value property.

## Is it possible to have a differentiable function on the real numbers?

It is possible to have a function defined for real numbers such that is a differentiable function everywhere on its domain but the derivative is not a continuous function . Equivalently, a differentiable function on the real numbers need not be a continuously differentiable function .