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.
Homework Equations
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...
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
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)?
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 [itex]\lim_{y\rightarrow0}g(x+y)=\lim_{y\rightarrow0}g(x)g(y)=g(x)g(0)=g(x)[/itex], that's about it.
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.
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.
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.
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.
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.