Epsilon-delta for a continuous function

Thread starter Nusc
Start date Nov 26, 2005
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Continuous Function

In summary, the epsilon-delta definition of continuity states that a function is continuous at a point if for every positive number ε, there exists a positive number δ such that for all x within δ units of c, the value of f(x) is within ε units of f(c). This definition captures the intuitive concept of continuity by ensuring that there are no sudden jumps or breaks in the function at the point c. It can be used to prove continuity by showing that for any given ε, there exists a δ that satisfies the definition. The choice of ε affects the choice of δ, with smaller values of ε requiring smaller values of δ. This definition can also be extended to multivariable functions by considering the distance between points in a multidimensional

Nov 26, 2005

Nusc

The function f is continuous at a E R.

Let f:D->R and D be a subset of R.
The function f is continuous at a E D if for every epsilon > 0 there exists a delta > 0 so that if |x-a|< delta and x E D then: |f(x)-f(a)|< epsilon.

Can someone check this for me?

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Nov 26, 2005

math-chick_41

that looks like the definition of continuity, is that all you wanted to know?

Nov 26, 2005

AKG

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Yes, that's right.

FAQ: Epsilon-delta for a continuous function

What is the definition of continuity in terms of epsilon-delta for a continuous function?

The epsilon-delta definition of continuity states that a function f is continuous at a point c if for every positive number ε, there exists a positive number δ such that for all x within δ units of c, the value of f(x) is within ε units of f(c).

How does the epsilon-delta definition of continuity relate to the intuitive concept of continuity?

The epsilon-delta definition of continuity captures the intuitive concept of continuity by requiring that the function does not have any sudden jumps or breaks at the point c. It ensures that as x gets closer and closer to c, the values of f(x) get closer and closer to f(c).

Can the epsilon-delta definition of continuity be used to prove that a function is continuous at a point?

Yes, the epsilon-delta definition of continuity can be used to prove that a function is continuous at a point by showing that for any given ε, there exists a δ that satisfies the definition. This is commonly done in mathematical proofs and is a rigorous way of showing continuity.

How does the choice of ε affect the choice of δ in the epsilon-delta definition of continuity?

The choice of ε affects the choice of δ because the smaller the value of ε, the smaller the range of values that f(x) has to be within in order to satisfy the definition. Therefore, a smaller ε requires a smaller δ to ensure continuity.

Can the epsilon-delta definition of continuity be extended to multivariable functions?

Yes, the epsilon-delta definition of continuity can be extended to multivariable functions by considering the distance between points in a multidimensional space instead of just on a single number line. This is known as the epsilon-delta definition of continuity in multivariable calculus.