# How do you prove something is differentiable and continuous?

Table of Contents

- 1 How do you prove something is differentiable and continuous?
- 2 Is the given statement true or false justify your answer if the function is continuous at then is differentiable at?
- 3 How do you prove a function is not continuous at a point?
- 4 What is the relationship between continuity and differentiability?
- 5 Are all differentiable functions on your are continuous on R?

## How do you prove something is differentiable and 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).)

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

Simply put, differentiable means the derivative exists at every point in its domain. Thus, a differentiable function is also a continuous function. But just because a function is continuous doesn’t mean its derivative (i.e., slope of the line tangent) is defined everywhere in the domain.

### Is the given statement true or false justify your answer if the function is continuous at then is differentiable at?

If a function is differentiable at a point then it must be continuous at that point. However, the converse need not be true. If a function is continuous at a point then it may or may not be differentiable at that point.

**How do you show a function is not differentiable at a point?**

A function is not differentiable at a if its graph has a vertical tangent line at a. The tangent line to the curve becomes steeper as x approaches a until it becomes a vertical line. Since the slope of a vertical line is undefined, the function is not differentiable in this case.

## How do you prove a function is not continuous at a point?

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. First x=−2 x = − 2 . The function value and the limit aren’t the same and so the function is not continuous at this point.

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

Differentiability Implies Continuity If is a differentiable function at , then is continuous at . Since we apply the Difference Law to the left hand side and use continuity of a constant to obtain that Next, we add on both sides and get that Now we see that , and so is continuous at .

### What is the relationship between continuity and differentiability?

We see that if a function is differentiable at a point, then it must be continuous at that point. There are connections between continuity and differentiability. Differentiability Implies ContinuityIf is a differentiable function at , then is continuous at .

**Is the function in figure a differentiable in Figure B?**

The function in figure A is not continuous at a, and, therefore, it is not differentiable there. In figures B – D the functions are continuous at a, but in each case the limit lim x → a f ( x) − f ( a) x − a does not exist, for a different reason. In figure B lim x → a + f ( x) − f ( a) x − a ≠ lim x → a − f ( x) − f ( a) x − a.

## Are all differentiable functions on your are continuous on R?

This theorem is often written as its contrapositive: If f ( x) is not continuous at x = a, then f ( x) is not differentiable at x = a. Thus from the theorem above, we see that all differentiable functions on R are continuous on R.