Are Continuous and Differentiable Functions on [0,1] Closed Under Operations?

[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.