## Is there a function that is differentiable but not 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.

## Are Holder continuous functions continuous?

For α > 1, any α–Hölder continuous function on [0, 1] (or any interval) is a constant. There are examples of uniformly continuous functions that are not α–Hölder continuous for any α.

**Can a function be not continuous?**

In other words, a function is continuous if its graph has no holes or breaks in it. For many functions it’s easy to determine where it won’t be continuous. Functions won’t be continuous where we have things like division by zero or logarithms of zero.

### Does differentiable imply 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.

### What is Equicontinuous family function?

In mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein. In particular, the concept applies to countable families, and thus sequences of functions.

**What has limit but not continuous?**

When a function is not continuous at a point, then we can say it is discontinuous at that point. There are several types of behaviors that lead to discontinuities. A removable discontinuity exists when the limit of the function exists, but one or both of the other two conditions is not met.

## Can a function have a limit but not be continuous?

Yes. Sin x is continuous, but the limit of sin x as doesn’t exist. Continuity at a point is the property that the limit of a function exists at that point and equals the value of the function there. A function is continuous if it is continuous at all of its points in its domain.

## Why is a differentiable function always continuous?

Simply put, differentiable means the derivative exists at every point in its domain. Consequently, the only way for the derivative to exist is if the function also exists (i.e., is continuous) on its domain. Thus, a differentiable function is also a continuous function.

**How do you prove that if a function is differentiable then it is continuous?**

Page 1

- Differentiable 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 prove a function is continuous?

1 Answer

- let p=1/α
- observe that |f′|p is an integrable function on (0,1)
- estimate |f(x)−f(y)| by the integral of |f′| over the interval between x and y.
- apply Hölder’s inequality to |f′|⋅1, raising |f′| to power p. This will yield a factor of |x−y|1/p as required.

### What is difference between equicontinuous and continuous?

As adjectives the difference between continuous and equicontinuous. is that continuous is without break, cessation, or interruption; without intervening time while equicontinuous is (mathematics|of a family of functions) such that all members are continuous, with equal variation in a given neighborhood.