Is a graph Continuous and differentiable at a given point

[betsinda]

Homework Statement

F
f(x)={(2x-1)/Absolute value(2x-1) x cannot equal (1/2)
{ 0 x = (1/2)

a) is f continuous at X = (1/2) explain
b) is f differentiable at x = (1/2) explain

Homework Equations

I have made the graph and x is a point at 1/2 but there is a jump. I have no idea how to start this.

The Attempt at a Solution

 

[Mark44]

Drawing a graph is a good start. Can you describe what the graph looks like?

 

[betsinda]

the line from the left approaches (1/2) at +1 and restarts at (1/2) at -1. There is a single point at (0,1/2)

 

[betsinda]

sorry it continues to the right at -1

 

[Mark44]

the line from the left approaches (1/2) at +1 and restarts at (1/2) at -1. There is a single point at (0,1/2)

No, none of this is right. The graph of this function is in three parts--two horizontal lines and a single point.

What is f(-1)? f(0)? f(1/2)? f(1)? f(2)? Plotting these points should give you an idea of what the graph of the function looks like.

 

[betsinda]

okay, I have fixed the graph. I have a line going from (1/2, 1) (1,1)(2,1) ect and a line going from (1/2,-1)(0,-1)(1,-1) ect, and a point at (1/2,0).

How do i determine if this is continuous at 1/2?

 

[Mark44]

One quibble. The line doesn't contain the point (1/2, -1). Does the graph look continuous at x = 1/2?

 

[betsinda]

I would say that the graph is not continuous at x=1/2 as that point does not connect to any other point on the graph.

 

[Mark44]

Correct.

 

[betsinda]

would it then be correct to say that x=1/2 is not differentiable as it is not continuous at that point?

 

[betsinda]

based on the fact that well every function is not differentiable, very function that is differentiable is continous. Or am I misunderstanding the concept?

 

[Mark44]

would it then be correct to say that x=1/2 is not differentiable as it is not continuous at that point?

No. It doesn't make any sense to talk about a point or an x value being differentiable. You can say, though, that a function is continuous or differentiable at a point or at some x value.

 

[Mark44]

based on the fact that well every function is not differentiable, very function that is differentiable is continous. Or am I misunderstanding the concept?

That is correct, so you are not misunderstanding the concept. I have made a couple of edits to what you wrote:

While not every function is differentiable, every function that is differentiable is continous.

 

[betsinda]

B) is f differentiable at x = (1/2) explain

So I could say that function is not differentiable at x=1/2 as the function is not not continuous?

Sorry for being a little slow I'm just trying to wrap my head around the concept. Thanks