Continuity of Functions at a Point: The Role of Addition and Multiplication

In summary, if the sum and difference of two functions f and g are continuous at x = x0, it does not necessarily mean that f and g themselves are continuous at x = x0. This is shown by the fact that if f is continuous and g is discontinuous at x = x0, then f + g may still be continuous at x = x0. However, the product of f and g may be discontinuous at x = x0, even if f is continuous at x = x0 and g is discontinuous at x = x0. This can be understood by using the equations f(x) = ((f + g)(x) + (f - g)(x))/2 and g(x) = ((
  • #1
dannysaf
10
0
1)Let f and g be functions such that f (x) + g(x) and f (x) − g(x) are
continuous at x = x0 . Must f and g be continuous at x = x0 ?

2)What can be said about the continuity of f (x) + g(x) at x = x0 , if
f (x) is continuous and g(x) is discontinuous at x = x0 ?

3)What can be said about the continuity of f (x)g(x) at x = x0 , if
f (x) is continuous and g(x) is discontinuous at x = x0 ?
 
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  • #2


I think the facts that f(x)= ((f+g)(x)+ (f-g)(x))/2 and g(x)= ((f+g)(x)- (f-g)(x))/2 will help a lot!
 

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

The continuity of a function means that the function is unbroken or uninterrupted throughout its domain. In other words, there are no gaps or jumps in the graph of the function.

2. How do you test for continuity of a function?

There are three main criteria for testing continuity of a function: 1) the function must be defined at the point of interest, 2) the limit of the function as x approaches the point must exist, and 3) the limit must equal the value of the function at that point.

3. Can a function be continuous at a certain point but not on its entire domain?

Yes, a function can be continuous at a certain point but not on its entire domain. This occurs when the function meets the first two criteria for continuity, but does not meet the third criterion, meaning the limit does not equal the value of the function at that point.

4. What is the difference between point-wise continuity and uniform continuity?

Point-wise continuity refers to the continuity of a function at individual points, while uniform continuity refers to the continuity of a function across its entire domain. A function that is point-wise continuous may not be uniformly continuous if it has a sharp change or jump at a certain point.

5. How does continuity of a function relate to differentiability?

A function is differentiable at a point if it is continuous at that point. However, a function can be continuous at a point but not differentiable, meaning it has a sharp turn or corner at that point. So, continuity is a necessary but not sufficient condition for differentiability.

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