# Combinations of Continuous Functions

• kingstrick
In summary, the homework statement is that if g is continuous at x=0 then it is continuous at every point of ℝ.
kingstrick

## Homework Statement

Let g: ℝ→ℝ satisfy the relation g (x+y) = g(x)g(y) for all x, y in ℝ. if g is continuous at x =0 then g is continuous at every point of ℝ.

## The Attempt at a Solution

Let W be an ε-neighborhood of g(0). Since g is continuous at 0, there is a δ-neighborhood V of 0 = f(c)...not sure where to proceed from here.

you want to prove lim_{y->0}g(x+y)=g(x), now you know how to use your condition on g.

Okay, so continuing with my proof:

At x = 0, then g(0+y) = g(o)g(y) = g(0)+g(y)
→ g(0) = g(y) (g(0)-1) ... then what? I don't understand how to proceed...

sunjin09 said:
you want to prove lim_{y->0}g(x+y)=g(x), now you know how to use your condition on g.

kingstrick said:

## Homework Statement

Let g: ℝ→ℝ satisfy the relation g (x+y) = g(x)g(y) for all x, y in ℝ. if g is continuous at x =0 then g is continuous at every point of ℝ.

## The Attempt at a Solution

Let W be an ε-neighborhood of g(0). Since g is continuous at 0, there is a δ-neighborhood V of 0 = f(c)...not sure where to proceed from here.

kingstrick said:
Okay, so continuing with my proof:

At x = 0, then g(0+y) = g(o)g(y) = g(0)+g(y)
→ g(0) = g(y) (g(0)-1) ... then what? I don't understand how to proceed...

First, check your derivation here for mistakes, and find what value g(0) may be. THEN use the hint I gave you for ARBITRARY x and y→0

sunjin09 said:
First, check your derivation here for mistakes, and find what value g(0) may be. THEN use the hint I gave you for ARBITRARY x and y→0

so am i missing a concept, when a problem says x→0 for the lim g, does that mean to apply to g(x) and g(y) not just g(x)?

kingstrick said:
so am i missing a concept, when a problem says x→0 for the lim g, does that mean to apply to g(x) and g(y) not just g(x)?

I just don't see how to determine the value of g(o)...I am stumped!

so is g(0) = 1?

You got it right. Since g(x)=g(x+0)=g(x)g(0), you have either g(0)=1 ( or g(x)=0 for all x's which is trivial.)
Now you can try to prove continuity by proving $\lim_{y\rightarrow0}g(x+y)=\lim_{y\rightarrow0}g(x)g(y)=g(x)g(0)=g(x)$, that's about it.

sunjin09 said:
You got it right. Since g(x)=g(x+0)=g(x)g(0), you have either g(0)=1 ( or g(x)=0 for all x's which is trivial.)
Now you can try to prove continuity by proving $\lim_{y\rightarrow0}g(x+y)=\lim_{y\rightarrow0}g(x)g(y)=g(x)g(0)=g(x)$, that's about it.

Thank you so much,i finally understand.

## 1. What is a combination of continuous functions?

A combination of continuous functions is a mathematical expression that combines two or more continuous functions using arithmetic operations such as addition, subtraction, multiplication, or division. It is also known as a composite function.

## 2. How is a combination of continuous functions different from a single continuous function?

A combination of continuous functions is different from a single continuous function in that it is made up of multiple functions that are connected together, whereas a single continuous function is just one function that remains continuous over its entire domain.

## 3. Can a combination of continuous functions be discontinuous?

Yes, a combination of continuous functions can be discontinuous if the functions used in the combination have points of discontinuity or if the combination results in a point of discontinuity. However, if all the functions used in the combination are continuous, then the combination will also be continuous.

## 4. What are some examples of combinations of continuous functions?

Some examples of combinations of continuous functions include polynomial functions, trigonometric functions, exponential functions, and logarithmic functions. For instance, f(x) = sin(x) + x^2 is a combination of the continuous functions sin(x) and x^2.

## 5. How are combinations of continuous functions useful in real-world applications?

Combinations of continuous functions are useful in real-world applications because they allow us to model complex relationships between different variables. They are frequently used in physics, engineering, economics, and other fields to describe real-world phenomena and make predictions based on the combination of multiple functions.

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