Is f(a+b) = f(a)f(b) true for all real numbers a and b?

Since we know that f is continuous at 0, we can replace a with 0 in the above equation to get:\lim_{x \to 0} f(x) = \lim_{x \to 0} f( (x - 0) + 0 ).Now, using the given information that f(a+b) = f(a)f(b), we can rewrite the right side as:\lim_{x \to 0} f(x) = \lim_{x \to 0} f(x) f(0).Since we already know that f(x) is continuous at 0, we can say that \lim_{x \to 0} f(x) = f(0). Therefore
  • #1
teme6@yahoo.c
1
0
there is one problem. the problem is related with contuinity of afunction and i tried like as shown below.so if anyone who is intersted to help me i like ..
the problem is
prove that if f(a+b)=f(a)f(b) for all a and b ,then f is cntiniuous at every real number.here there is given information that is the domain of f is the set of all real number and f is continuous at 0.
I tried to approache this problem like this
since f is continuous at 0, f(0) must defined and it must be equal to the value of limf(x) =f(0)
x-->0
and a and b are real numbers then limf(a+b)=limf(a).limf(b)
x--->0 x-->0 x-->0
=f(a).f(b) (since f(a) and f(b) are constnt fu.)
this implies f is continuous at every real number.
am i write? i want to be sure so please help
 
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  • #2
Hi, welcome to PF.

Try looking at
[tex]\lim_{x \to a} f(x) = \lim_{x \to a} f( (x - a) + a )[/tex].
 

Related to Is f(a+b) = f(a)f(b) true for all real numbers a and b?

1. What is the definition of continuity for a function?

The continuity of a function is defined as the property of a function where the graph of the function has no breaks or jumps, and the function is defined at every point in its domain.

2. How can you determine if a function is continuous at a specific point?

A function is continuous at a specific point if the limit of the function as x approaches that point is equal to the value of the function at that point.

3. Can a function be continuous at a point but not on an interval?

Yes, a function can be continuous at a point but not on an interval if it has a discontinuity within that interval.

4. What are the three types of discontinuities that a function can have?

The three types of discontinuities are removable, jump, and infinite. A removable discontinuity occurs when there is a hole in the graph of the function. A jump discontinuity occurs when there is a sudden jump or gap in the graph. An infinite discontinuity occurs when the function approaches positive or negative infinity at a certain point.

5. What is the difference between a continuous and a differentiable function?

A continuous function is one that has no breaks or jumps in its graph, while a differentiable function is one that is continuous and also has a well-defined derivative at every point in its domain. In other words, a differentiable function is a specific type of continuous function that is also "smooth" with no sharp turns or corners.

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