Is the Limit Definition of Continuity Equivalent to the Standard Definition?

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In summary, continuity is expressed as both lim x->a f(x)=f(a) and lim h->0 f(x+h) - f(x) = 0. This is because both expressions imply that all numbers around f(x) are very close to f(x), which is the essence of continuity.
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Freye
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Hey guys,

Continuity is generally expressed as lim x->a f(x)=f(a).
But is it also correct to express it as: lim h->0 f(x+h) - f(x) = 0?
Because that would imply that all numbers around f(x) would have to be very close to f(x), and that is basically what continuity is, no?
 
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  • #2
Freye said:
Hey guys,

Continuity is generally expressed as lim x->a f(x)=f(a).
But is it also correct to express it as: lim h->0 f(x+h) - f(x) = 0?
Because that would imply that all numbers around f(x) would have to be very close to f(x), and that is basically what continuity is, no?

Yes, this is fine (assuming f(x) exists)
 
  • #3
Ok thank you, it really helps me out on a problem I am working on.
 

Related to Is the Limit Definition of Continuity Equivalent to the Standard Definition?

1. What is continuity and why is it important in science?

Continuity refers to the unbroken and consistent flow of a phenomenon or process. In science, it is important because it allows us to make accurate predictions and understand the behavior of systems over time. It also helps us to identify any disruptions or changes in a process.

2. What are some ways to express continuity in scientific data?

One way to express continuity is through graphs and charts that show the trend or pattern of a process over time. Another way is through mathematical equations that describe the relationship between variables. Additionally, continuity can be expressed through written descriptions or diagrams.

3. How does continuity differ from stability?

While continuity refers to the consistent flow of a process, stability refers to the ability of a system to resist changes and return to its original state. A system can be both continuous and stable, but continuity does not guarantee stability. For example, a system may continue to change but still be stable in its new state.

4. Can continuity be broken or disrupted?

Yes, continuity can be broken or disrupted by various factors such as external forces, changes in the environment, or errors in data collection. In some cases, the continuity may be re-established after the disruption, while in others it may lead to a completely new process.

5. How can we ensure continuity in scientific experiments or observations?

To ensure continuity, it is important to carefully plan and execute experiments or observations while minimizing any external factors that could cause disruptions. It is also crucial to use accurate and reliable methods of data collection and analysis to maintain the integrity of the process.

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