What is the difference between continuity and uniform continuity of functions The expert replied that uniform continuity is more strict than continuity. A function that is uniformly continuous on an interval is continuous, but a continuous function is not uniformly continuous. But the theorem in the book clearly says that if a function is continuous in a closed interval, it is uniformly continuous in that interval. Which is more strict? Is there a continuous function that is not uniformly continuous?

What is the difference between continuity and uniform continuity of functions The expert replied that uniform continuity is more strict than continuity. A function that is uniformly continuous on an interval is continuous, but a continuous function is not uniformly continuous. But the theorem in the book clearly says that if a function is continuous in a closed interval, it is uniformly continuous in that interval. Which is more strict? Is there a continuous function that is not uniformly continuous?

You are right. Continuous functions are indeed uniformly continuous in closed intervals, but not necessarily in open intervals. The definition of continuous functions is that every point is continuous, while for the same epsilon > 0, the corresponding delta of each point is different. But uniform continuity requires a certain delta to meet all points, so it is more strict

What is the difference between "continuity" and "uniform continuity" of functions in freshman mathematical analysis? RT

Continuity is a local property. Generally, it is only discussed for a single point. It is said that the continuity of a function on a set is only point by point
Uniform continuity is a global property. It is necessary to discuss a subset (such as interval) on the definition domain to show the continuity of the whole
Consistent continuity can deduce continuity, otherwise
This must be made clear, otherwise you won't understand it when you learn uniform convergence and future equicontinuity and absolute continuity

Proof of function continuity Given that f (x) and G (x) are continuous at x0, it is proved that h (x) = max (f (x), G (x)) is continuous at x0

First, if f (x) is continuous at a certain point, it is easy to prove that f (x) is also continuous at that point
And H (x) = (f (x) + G (x) + |f (x) - G (x) | / 2
So h (x) is continuous at x0

A proof of function continuity If f (x) is continuous at x = 0 and f (x + y) = f (x) + F (y), it is true for any x, y ∈ (- infinity, + infinity). Try to prove that f (x) is a continuous function on (- infinity, + infinity)

When x = 0, f (y) = f (0) + F (y)
Then f (0) = 0
Since f (x) is continuous at x = 0, there is f (x) - > 0 (X -- > 0)
For any
f(x+ Δ x)-f(x)=f( Δ x) -- > 0 when Δ x-->0
So the continuity of F (x) is proved

How to prove the continuity of function?

1、 If the function is known to be an elementary function, it means that it is an elementary function and is continuous in its definition interval;
2、 If the function is a univariate function, the derivative of the function can be obtained. If the derivative is meaningful at a certain point, the point must be continuous --- the derivative must be continuous;
3、 If you can't, you have to find the limit. If the limit of the function at that point is equal to the value of the function at that point, it is continuous;
Note: the left and right limits are only a part of the limit calculation. When the function is a piecewise function, the limit calculation method at the piecewise point must use the left and right limits

Proof of function continuity It's the problem on the picture. Don't use the proof method on calculus. It's best to use the original image of F continuous, that is, open set, to prove it

x. To what extent is y discussed? I think of it as a topological group ~ it should be broad enough
According to the definition of continuity in topology - the original image of open set is open set, it is easy to prove that the composite function of continuous function is still continuous
H (x, y) = u (x, y ^ (- 1)) = u (x, V (y)). Since u and V are continuous, h is continuous