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In summary, "Epsilontic – Limits and Continuity" explores the foundational concepts of limits and continuity in calculus using the epsilon-delta definition. It emphasizes the precise criteria for a function to approach a certain value (limit) as inputs get closer to a specific point, and outlines the conditions under which a function is considered continuous. The text illustrates these concepts through examples and theorems, highlighting their significance in understanding mathematical analysis and real-world applications.
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nuuskur
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I remember it took me a while to realise if a nonnegative number is smaller than any positive number, then it would have to be zero. Then I started understanding sandwiching arguments of the form
[tex]
0\leqslant |f(x)-f(a)| \leqslant g(x) \xrightarrow[x\to a]{}0.
[/tex]
It's so obvious in retrospect, how on earth could I not understand something this simple..?
[tex]
0\leqslant |f(x)-f(a)| \leqslant g(x) \xrightarrow[x\to a]{}0.
[/tex]
It's so obvious in retrospect, how on earth could I not understand something this simple..?
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I think there is a general difficulty that students have to overcome when they take the step from school to university. Perspective changes from algorithmic solutions toward proofs, techniques are new, and all of it is explained at a much higher speed, if at all since you can read it by yourself in a book, no repetitions, no algorithms. There are so many new impressions and rituals that it is hard to keep up. I still draw this picture of a discontinuous function in the article sometimes to sort out the qualifiers in the definition when I want to make sure to make no mistake: which comes first and which is variable, which depends on which.nuuskur said:I remember it took me a while to realise if a nonnegative number is smaller than any positive number, then it would have to be zero. Then I started understanding sandwiching arguments of the form
[tex]
0\leqslant |f(x)-f(a)| \leqslant g(x) \xrightarrow[x\to a]{}0.
[/tex]
It's so obvious in retrospect, how on earth could I not understand something this simple..?
FAQ: Epsilontic – Limits and Continuity
What is the epsilon-delta definition of a limit?
The epsilon-delta definition of a limit formalizes the concept of limits in calculus. It states that the limit of a function f(x) as x approaches a value c is L (written as lim (x → c) f(x) = L) if for every positive number ε (epsilon), there exists a positive number δ (delta) such that whenever 0 < |x - c| < δ, it follows that |f(x) - L| < ε. This means that we can make the values of f(x) arbitrarily close to L by choosing x sufficiently close to c.
How do you prove a limit using the epsilon-delta definition?
To prove a limit using the epsilon-delta definition, you need to show that for a given ε > 0, you can find a δ > 0 such that the condition |f(x) - L| < ε holds whenever 0 < |x - c| < δ. This usually involves manipulating the inequality |f(x) - L| < ε to express it in terms of |x - c|, allowing you to find an appropriate δ that satisfies the condition. The proof is often done by algebraic manipulation and sometimes requires bounding certain expressions.
What is continuity in terms of limits?
A function f(x) is said to be continuous at a point c if the following three conditions are met: 1) f(c) is defined, 2) the limit of f(x) as x approaches c exists, and 3) the limit of f(x) as x approaches c is equal to f(c). In other words, the function does not have any jumps, breaks, or holes at that point, allowing for a smooth transition as x approaches c.
What are the implications of discontinuity?
Discontinuity implies that there is a break in the behavior of the function at a certain point. This can lead to various implications such as the function not being integrable over an interval that includes the discontinuity, difficulties in applying the Fundamental Theorem of Calculus, and challenges in optimization problems. Discontinuous functions can also exhibit unpredictable behavior, making them more complex to analyze and work with mathematically.
Can a function be continuous everywhere but not differentiable at some points?
Yes, a function can be continuous everywhere but not differentiable at some points. A classic example is the absolute value function, f(x) = |x|, which is continuous at all points but has a sharp corner at x = 0, where it is not differentiable. Continuity at a point means there are no breaks in the function, while differentiability requires that the function has a well-defined tangent (slope) at that