I'm having trouble with a certain math problem. Here is the exact problem:
Let f, g: ℝ -> ℝ be continuous functions.
Prove: For all α > 0 and β > 0, the function F: ℝ -> ℝ defined by F(x) = α * f(x) + β * g(x) is continuous.
I am quite certain that I have to solve this using the epsilon delta definition. I have no problems proving a function like "f(x) = 7x2 + 5x + 8" is continuous using the epsilon delta definition, but I don't know how to approach this one because the functions f(x) and g(x) aren't formulas like that one.
I understand that the epsilon delta definition means that if the R ↦ R is continuous in x = p, then there exists a δ > 0 for every ε > 0 such that it follows from | x - p | < δ that | f(x) - f(p) | < ε.
It seems to me that in this case, I need to find the δ for which δ > |p - x| implies that ε > |f(p)α-f(x)α|, where it is given that ε > 0 and δ > 0.
And then I get stuck, because I wouldn't know how to find that δ. I can't manage to get any further than |αf(x)-αf(p)|<|αf(p)-αf(p+δ)|.
How should I approach this? Thanks.
Let f, g: ℝ -> ℝ be continuous functions.
Prove: For all α > 0 and β > 0, the function F: ℝ -> ℝ defined by F(x) = α * f(x) + β * g(x) is continuous.
I am quite certain that I have to solve this using the epsilon delta definition. I have no problems proving a function like "f(x) = 7x2 + 5x + 8" is continuous using the epsilon delta definition, but I don't know how to approach this one because the functions f(x) and g(x) aren't formulas like that one.
I understand that the epsilon delta definition means that if the R ↦ R is continuous in x = p, then there exists a δ > 0 for every ε > 0 such that it follows from | x - p | < δ that | f(x) - f(p) | < ε.
It seems to me that in this case, I need to find the δ for which δ > |p - x| implies that ε > |f(p)α-f(x)α|, where it is given that ε > 0 and δ > 0.
And then I get stuck, because I wouldn't know how to find that δ. I can't manage to get any further than |αf(x)-αf(p)|<|αf(p)-αf(p+δ)|.
How should I approach this? Thanks.