Mathematical Analysis - Continuity

In summary, Continuity is a mathematical concept that describes the smoothness or connectedness of a function. It is defined as a function being continuous at a point if the limit of the function exists and is equal to the function's value at that point. The continuity theorem states that if two functions are continuous at a point, their sum, difference, product, and quotient (as long as the denominator is not 0) are also continuous at that point. To determine if a function is continuous, it must satisfy three conditions: it is defined at the point, the left and right-hand limits exist and are equal, and the function's value at that point is equal to the limit. Continuity is important in mathematics because it is essential for understanding
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rollsroy
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Homework Statement



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Homework Equations


The Attempt at a Solution



Basically the first part of the question asks to prove the binomial theorem through induction which I've done.

I'm basically lost as to how to even attempt these questions, I'm not asking for answers as I need to know this stuff for exams and other courses.

Can anyone point me to a place where I can find similar style of questions with an explanation on the theory on how to do them?

We've only been shown in lectures on how to show continuity through epsilon-delta arguments?
 

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FAQ: Mathematical Analysis - Continuity

What is continuity?

Continuity is a mathematical concept that describes the smoothness or connectedness of a function. A function is said to be continuous if there are no abrupt changes or gaps in its graph.

How is continuity defined?

Continuity is defined as a function f(x) being continuous at a point x = a if the limit of f(x) as x approaches a exists and is equal to f(a). In other words, the function's value at a point is equal to the limit of its values as the input approaches that point.

What is the continuity theorem?

The continuity theorem states that if two functions, f(x) and g(x), are continuous at a point x = a, then the sum, difference, product, and quotient of these functions (as long as g(a) ≠ 0) are also continuous at x = a.

How can we determine if a function is continuous?

A function is continuous if it satisfies the three conditions of continuity: 1) the function is defined at the point in question, 2) the left-hand limit and right-hand limit exist at that point and are equal, and 3) the function's value at that point is equal to the limit.

What is the importance of continuity in mathematics?

Continuity is a fundamental concept in mathematics and is essential for understanding limits, derivatives, and integrals. It also allows us to make approximations and predictions about a function's behavior at points where it is not defined.

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