(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

f(x) = 4 for x > or = 0, f(x) = 0 for x < 0, and g(x) = x^2 for all x.

Thus dom(f) = dom(g) = R.

2. Relevant equations

a. Determine the following functions: f+g, fg, f o g, g o f. Be sure to specify thier domain.

b. Which of the functions f, g, f+g, fg, f o g, g o f is continuous

3. The attempt at a solution

Ok, so for part (a) I am at f o g step, which I say is f(x^2) = 4 for x > or = 0 and f(x^2) = 0 for x < 0.

Question 1. Since f is a function of g(x), then when I restrict it values, do I have to restrict them in terms of x (as given) or in terms of x^2, since it is a f of g(x) now? Would I have to say that f(x^2) = 4 for x^2 > or = 0 and f(x^2) = 0 for x^2 < 0, instead of how I provided above? If I do have to say it in terms of x^2, then x^2 < 0 makes no sense, and the f would not be defined as f(g(x)) < 0?

Question 2. Since the composition f o g from Q1 is not piecewise anymore and starts at 0 and goes to positive infinity, would it make the composition continuous function?

Question 3. I am a little confused on how to prove that f is not a continuous function using limits. Would someone be able to show me a quick example please?

Question 4. If f is not continuous, then I assume that f, g, f+g, fg are not continuous either. But if f o g IS continuous (Q3), then wouldnt it contradict a Theorem that states "If f is continuous at x0 and g is continuous at f(x0), then the composite functions g o f is continuous at x0"?

Thank you for your time.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Continuity (real analysis)

**Physics Forums | Science Articles, Homework Help, Discussion**