Constant Functions: ε-δ Proof for Continuity

Thread starter Niles
Start date Jan 15, 2010
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Constant Functions

In summary, a constant function is a type of mathematical function where every output value is the same, regardless of the input value. To prove continuity for a constant function using ε-δ definition, we need to show that for any ε > 0, there exists a δ > 0 such that for all x within δ of the given input value, the output values do not deviate from the constant value by more than ε. A constant function cannot be discontinuous as it always has the same output value for any given input value. It differs from a linear function in that a linear function has a constant rate of change (slope) between the input and output values. The significance of using ε-δ definition for proving continuity is that it

Jan 15, 2010

Niles

Homework Statement

Hi all

How can I show that a constant function defined on a closed interval is continuous on that same interval using a ε-δ proof?

I have f(x)=c on the interval. Then I write

f(x)-f(a) = c-c = 0

for some point a in the interval. But what to do from here?

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Jan 15, 2010

Mark44

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Insights Author

This is a very easy proof. Isn't 0 < epsilon for any positive epsilon?

1. What is the definition of a constant function?

A constant function is a type of mathematical function where every output value is the same, regardless of the input value. This means that the graph of a constant function is a horizontal line.

2. How do you prove continuity for a constant function using ε-δ definition?

To prove continuity for a constant function using ε-δ definition, we need to show that for any ε > 0, there exists a δ > 0 such that for all x within δ of the given input value, the output values do not deviate from the constant value by more than ε.

3. Can a constant function be discontinuous?

No, a constant function cannot be discontinuous as it always has the same output value for any given input value. Therefore, it satisfies the definition of continuity at every point.

4. How does a constant function differ from a linear function?

A constant function has a constant output value regardless of the input value, while a linear function has a constant rate of change (slope) between the input and output values. In other words, a linear function is a straight line, while a constant function is a horizontal line.

5. What is the significance of using ε-δ definition for proving continuity?

The ε-δ definition allows us to mathematically prove the continuity of a function at a specific point. It provides a precise and rigorous way to show that a function is continuous at a given point, rather than relying on visual inspection or intuition.