frankpupu said:Homework Statement
function f:R->R can be written as a sum f=f1+f2 where f1 is even and f2 is odd。show that if f is continuous then f1 and f2 may be chosen continuous, and if f is differentiable then f1 and f2 can be chosen differentiable
2. The attempt at a solution
i have try some examples, but i still cannot get the idea from that
HallsofIvy said:Given function f, let [itex]f_e(x)= (f(x)+ f(-x))/2[/itex] and [itex]f_o(x)= (f(x)- f(-x))/2[/itex].
frankpupu said:yes f(x)=((f(x)+f(−x))/2 )+(f(x)−f(−x))/2) then (f(x)+f(−x))/2 is even and (f(x)−f(−x))/2 is odd,then i don't know how to argue their continuity and differentiability?
Dick said:I don't think you have to prove it from scratch. If f(x) is continuous then f(-x) is continuous. What theorem about continuous functions might you use to prove that?
frankpupu said:if f(x)is continuous then f(-x)is continuous ,(f(x)+f(-x))/2 is continuous by sum rule (f(x)-f(-x))/2 is continuous as well .but they should be chosen continuous
A continuous function is one that has no breaks or gaps in its graph, meaning that the values of the function change smoothly and continuously as the input changes. A differentiable function, on the other hand, is one that has a well-defined derivative at every point in its domain, meaning that it has a well-defined slope or rate of change at every point.
A function is continuous if it has no breaks or gaps in its graph and if the limit of the function as the input approaches a particular point is equal to the value of the function at that point. In other words, the function must have a smooth and unbroken graph, and there should be no sudden jumps or discontinuities in its values.
Differentiability is important because it allows us to calculate the rate of change of a function at any given point. This is useful in many real-world applications, such as calculating the velocity of an object or the growth rate of a population. It also allows us to find the maximum and minimum values of a function, which can be useful in optimization problems.
Yes, a function can be continuous but not differentiable. This means that the function has a smooth and unbroken graph, but it does not have a well-defined slope or rate of change at every point. This can happen, for example, when there is a sharp corner or cusp in the graph of the function.
A function is differentiable if it has a well-defined derivative at every point in its domain. To determine if a function is differentiable, you can use the definition of the derivative or check if the function satisfies the differentiability criteria, such as being continuous and having a well-defined limit at every point.