Continuous and Differentiable Functions on [0,1]: Exploring G and H Sets

[Ted123]

If G=the set of all continuous complex-valued functions on the interval [0,1] and $f,g \in G$ then is $$\displaystyle f(x) \int^1_0 g(t) \; dt - g(x) \int^1_0 f(t)\;dt$$ in G?

If H=the set of all differentiable complex-valued functions on the interval [0,1] and $f,g \in H$ then is $$fg' - gf'$$ in H?

 

[HallsofIvy]

$\int_0^1 f(x) dx$ and $\int_0^1 g(x)dx$ are numbers so, knowing that f and g are continuous, what can you say about Af+ Bg for constants B and G?

The derivative of f'g+ fg' is f''g+ 2f'g'+ fg''. Knowing that f and g are differentiable what can you say about f'' and g''?

 

[Ted123]

$\int_0^1 f(x) dx$ and $\int_0^1 g(x)dx$ are numbers so, knowing that f and g are continuous, what can you say about Af+ Bg for constants B and G?

The derivative of f'g+ fg' is f''g+ 2f'g'+ fg''. Knowing that f and g are differentiable what can you say about f'' and g''?

Got it. They're both still in the sets.