Monotonically decreasing function

In summary, If a function f:R->R is monotonically decreasing, then every discontinuous point of f has finite right and left limits which are unequal. Additionally, if every continuous function defined in a finite segment I, where a<b and a,b are in R, has a maximum and a minimum, then I is a closed segment [a,b].
  • #1
daniel_i_l
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Homework Statement


True or false:
1) If f:R->R is a monotonically decreasing function then every discontinues point of f has finate right and left limits which are unequal.
2) I is some finate segment where a<b and a,b in R. If every continues function defined in I has a maximum and a minimum then I is a closed (in other words [a,b]) segment.


Homework Equations





The Attempt at a Solution



1) I think this is true but how can I prove it? Can someone give me a push in the right direction?
2)True: f(x)=x is continues and defined in any segment but it only has a min and max in an closed one. Is that right?

Thanks.
 
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  • #2
2) looks fine. For 1) try thinking of it this way. Let c be the discontinuous point. For the left limit consider the sequence f(c-1/n) for integers n>=1. The sequence is decreasing and bounded below (by f(c)) so it has a limit L. Can you show R is the left hand limit?
 
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What is a monotonically decreasing function?

A monotonically decreasing function is a mathematical function whose values decrease as the input values increase. This means that the function is always moving downwards, from left to right, on a graph.

How is a monotonically decreasing function different from a monotonically increasing function?

A monotonically decreasing function and a monotonically increasing function are opposite types of mathematical functions. While a monotonically decreasing function decreases as the input values increase, a monotonically increasing function increases as the input values increase.

What is the difference between a strictly monotonically decreasing function and a non-strictly monotonically decreasing function?

A strictly monotonically decreasing function is a function that decreases without any flat intervals or plateaus, meaning the function is always moving downwards without any horizontal sections. A non-strictly monotonically decreasing function, on the other hand, may have some flat intervals or plateaus where the function does not decrease.

What are some real-life examples of monotonically decreasing functions?

A common example of a monotonically decreasing function is the value of a new car. As time passes and the car is used, its value decreases. Another example is the temperature of a cup of hot coffee, which decreases as it sits and cools down.

How can you determine if a function is monotonically decreasing?

To determine if a function is monotonically decreasing, you can graph the function and see if it always moves downwards from left to right. Another way is to calculate the derivative of the function and see if it is always negative, indicating a decreasing trend.

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