Continuous Function Proof

In summary, a continuous function is a type of mathematical function with a smooth and gradual change in output as the input changes. To prove continuity, the function must satisfy three conditions. Proving continuity is important for accurate predictions and understanding a function's behavior. Common techniques for proving continuity include using the definition, algebraic manipulations, and theorems. It is not necessary for all continuous functions to be differentiable.
  • #1

Homework Statement

Prove that if f is continuous at a, then so is |f|

Homework Equations

The Attempt at a Solution

I know
lim f = L

Not sure really where to go from here.
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  • #2
to prove that f is continuous at a, you must prove that [tex]f(a)[/tex] is defined, that [tex]\lim_{x \to a} f(x)[/tex] exists and that [tex]\lim_{x \to a} f(x) = f(a)[/tex].
  • #3
Thanks I should be ok from here.

1. What is a continuous function?

A continuous function is a type of mathematical function where the output value changes smoothly and gradually as the input value changes. This means that there are no sudden jumps or breaks in the function's graph.

2. How do you prove that a function is continuous?

To prove that a function is continuous, you must show that it satisfies the three conditions of continuity: 1) the function is defined at the point in question, 2) the limit of the function at that point exists, and 3) the limit of the function at that point is equal to the function's value at that point.

3. What is the importance of proving continuity?

Proving continuity is important because it allows us to make accurate predictions and calculations based on a function's behavior. It also helps us understand the behavior of a function and its relationship to other mathematical concepts.

4. What are some common techniques used to prove continuity?

Some common techniques used to prove continuity include using the definition of continuity, using algebraic manipulations, and using theorems such as the Intermediate Value Theorem and the Squeeze Theorem.

5. Are all continuous functions differentiable?

No, not all continuous functions are differentiable. While all differentiable functions are continuous, the opposite is not always true. A function can be continuous but not differentiable if it has a sharp point or corner in its graph, known as a point of non-differentiability.

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