W is the set of all functions that are differentiable on 11

W is the set of all functions that are differentiable on [-1,1]. V is the set of all functions that are continuous on [-1,1]. Is W a vector space of V.

Solution

If a function is differentiable, it is necessarily continuous.

But a continuous function need not be differentiable.

An example is f(x) = |x|

This funciton is continuous at x=0 as left limit = right limit = f(0)

But f(x) = |x| is not differentiable at x=0

since f(x) = -x, x<0

= x , x >0

f\'(x) = -1, x <0

= 1, x >=0

= 0 , x =0

Since these 3 are different f is not differentiable at x=0

Here W is the set of all differentiable functions.

V the set of continuous functions in [-1,1]

Since all differentiable functions must also be continuous, W is a subset of V.

Consider two differentiable functions f(x) and g(x) in W.

f(x)+g(x) is also differentiable and is in W.

f(x) and g(x) are in W implies f+g belongs to W.

Addition is closed.

f(x) =0 a differentiable funciton in [-1,1] is the identity for W

and f^-1(x) = -f(x) differentiable and hence in W

If f(x) is differentiable in [-1,1] then cf(x) and (c+d)f(x) are differentiable.

So W is a vector space.