- #1
dcl
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How do I show that f(x)=x^2 is continuous at any given point, say x=3.
Thank you.
Thank you.
dcl said:I'm sorry, I don't really understand the 'method' or the reasoning behind that. I need to learn it, but can't for the life of me understand it at the moment.
Where did the epsilon in the 3rd last line come from?
franznietzsche said:Actually the easiest way to prove continuity at all values is to show that the derivative is always defined, differentiability always implies continuity (note the converse is not always true.). So for f(x) = x^2, you get f'(x) = 2x, which is defined for all values of x thus f(x) is continuous across the interval (-infinity,infinity)
dcl said:I'm sorry, I don't really understand the 'method' or the reasoning behind that. I need to learn it, but can't for the life of me understand it at the moment.
Where did the epsilon in the 3rd last line come from?
matt grime said:*cough* where have you proved that the derivative exists and is equal to 2x? You are assuming x^2 is diffble, and in that case you might as well assume it is continuous, mightn't you?
franznietzsche said:umm...use the definition of the derivative as the ratio of [tex] \frac{\Delta f}{\Delta x} [/tex] and you will get that answer, always. So its not an assumption. Try the theorems of differential calculus.
franznietzsche said:umm...use the definition of the derivative as the ratio of [tex] \frac{\Delta f}{\Delta x} [/tex] and you will get that answer, always. So its not an assumption. Try the theorems of differential calculus.
Continuity at a point refers to a mathematical concept that describes the smoothness of a function at a specific point. It means that the function has no abrupt changes or breaks at that point.
To prove continuity at a point, you need to show that the limit of the function exists at that point, and it is equal to the value of the function at that point. This can be done using the epsilon-delta definition of continuity or by using the three-part definition of continuity.
The epsilon-delta definition of continuity states that a function f is continuous at a point x=a if for any given epsilon > 0, there exists a delta > 0 such that for all x within the interval (a-delta, a+delta), the difference between f(x) and f(a) is less than epsilon.
The three-part definition of continuity states that a function f is continuous at a point x=a if the following three conditions are met: 1) the function is defined at x=a, 2) the limit of the function as x approaches a exists, and 3) the limit is equal to the value of the function at x=a.
Proving continuity at a point is important because it ensures that the function is well-behaved and has no abrupt changes or breaks, making it easier to analyze and use in mathematical calculations. It also allows us to make conclusions about the behavior of the function at that point and in its neighborhood.