# What does continuity mean?

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Continuity refers to the property of a function or a curve that implies it has no abrupt changes or breaks. It means that the function or curve is smooth and connected without any gaps or jumps. In mathematical terms, a function is continuous if, for any given input, the output values approach each other as the input values get closer. Continuity is an essential concept in calculus and analysis, as it allows for the study of limits, derivatives, and integrals.

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Continuity refers to a property of functions in mathematics. A function is continuous if it has no breaks, jumps, or holes in its graph. In other words, a function is continuous at a point if the value of the function approaches the same value from both sides of that point as the input approaches that point. This means that there are no sudden changes or disruptions in the function's behavior.

Mathematically, a function ( f(x) ) is continuous at a point ( x = a ) if three conditions are met:

- ( f(a) ) is defined (i.e., the function is defined at ( x = a )).
- The limit of the function as ( x ) approaches ( a ) exists.
- The limit of the function as ( x ) approaches ( a ) is equal to the value of the function at ( x = a ).

If these conditions are satisfied for every point in the domain of the function, then the function is considered continuous over its entire domain. Continuity is a fundamental concept in calculus and real analysis, as it allows mathematicians to study the behavior of functions and make predictions about their values and properties.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- How do you find values of δ that correspond to ε=0.1, ε=0.05, and ε=.01 when finding the limit of #(5x-7)# as x approaches 2?
- What is the limit of #[x^2+3x+4]/[x^3-3x^2+1] + [x^4-x]/[x^5]# as x goes to infinity?
- How do you evaluate the limit #(2-cos^2x)/x^2# as x approaches #0#?
- How do i find discontinuity for a function?
- What is the limit of #((5u^4)+8)/((u^2)-5)((7u^2)-1)# as x goes to infinity?

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