# Evaluate where F(x) is differentiable

1. ### sprite1608

2
Hi there, I cannot seem to figure this question out.

1. The problem statement, all variables and given/known data
Let f: [0,3] -> R be defined as follows

x if 0≤x<1,​
f(X)= 1≤x<2
x if 2≤x≤3​

obtain formulas for F(x) = for 0≤x≤3 and sketch the graphs of f(x) and F(x). Where is F(x) differentiable? Evaluate F(x) where differentiable.

I missed a day of class and am now totally lost. I've read through the sections in Intro to real analysis by Bartle that cover this section and I get no where. Any help on where to go would be greatly appreciated!

2. ### HallsofIvy

40,939
Staff Emeritus
You are missing the definition of f(x) for x between 1 and 2.

Hopefully, you know that f(x)= x is differentiable for all x. I suspect that the missing formula for x between 1 and 2 also defines a differentiable function. If that is the case, the only problem is whether the function is differentiable at the "joints", x= 1 and x= 2. Apply the definition of the derivative,
$$\lim_{h\to 0}\frac{f(a+h)- f(a)}{h}$$
with a= 1 and then with a= 2. Look at the one sided limits.

3. ### sprite1608

2
shoot. I missed that when I checked it over. it is f(x) = 1 if 1≤x<2

With that information, would it really make much of a difference to what you said previously?

4. ### SammyS

9,036
Staff Emeritus
You have f(x) and F(x). What relationship is being assumed between those two functions?