Proof of Continuity: Homework Statement

In summary, the conversation discusses whether the functions f and g are continuous if the function f+g is continuous. A proof is presented using a piecewise function to show that it is possible for f and g to be discontinuous even if f+g is continuous, thus disproving the given statement.
  • #1
mrchris
31
0

Homework Statement


If the function f+g:ℝ→ℝ is continuous, then the functions f:ℝ→ℝ and g:ℝ→ℝ also are continuous.


Homework Equations





The Attempt at a Solution


Ok, just learning my proofs here, so I'm not sure if my solution is cheating or not rigorous enough. take f(x)= {-1 if x≥0, 1 if x<0} and take g(x)= {1 if x≥0, -1 if x<0}. Then the function (f+g)(x) is a constant function equal to 0 everywhere. since g(x) and f(x) are both discontinuous at x=0, this is a contradiction to the given statement. Basically, I don't know if it is ok to use a piecewise function like this to disprove a statement.

 
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  • #2
Yes. That's perfect. You've found discontinuous functions f and g such that f+g is continuous. So the statement is false.
 

FAQ: Proof of Continuity: Homework Statement

What is proof of continuity?

Proof of continuity is a mathematical concept that shows that a function is continuous over a particular interval or domain. It involves using the definition of continuity to show that a function has no breaks or gaps in its graph.

How do you prove continuity using the limit definition?

To prove continuity using the limit definition, you need to show that the limit of the function as x approaches a particular value is equal to the value of the function at that point. In other words, the left-hand limit and the right-hand limit must both exist and be equal at the point in question.

What are some common strategies for proving continuity?

Some common strategies for proving continuity include using the definition of continuity, using the limit definition, and using algebraic manipulation to simplify the function. Additionally, it can be helpful to use the properties of continuous functions such as the sum, difference, product, and quotient rules.

How does continuity relate to differentiability?

Continuity is a necessary but not sufficient condition for differentiability. This means that a function must be continuous in order to be differentiable, but being continuous does not guarantee differentiability. A function can be continuous without having a derivative at a particular point if it is not differentiable at that point.

Can a function be continuous but not differentiable?

Yes, a function can be continuous but not differentiable. This can occur at points where there is a sharp turn or corner in the graph, or where there is a vertical tangent line. In these cases, the function is still continuous, but the derivative does not exist at that point.

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