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  1. JanGullberg
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    2023-01-12T11:27:23+00:00
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    If F Is Continuous On (−∞, ∞), What Can You Say About Its Graph?

    Introduction

    Calculus is a powerful tool in mathematics, and one of its fundamentals is the concept of a continuous function. A continuous function is one that can be drawn in one stroke without lifting the pencil from the paper, meaning that no matter how small or large we zoom in, the graph still looks smooth. But what does this mean for functions with an infinite domain? How do these graphs look, and what can we say about them? In this blog post, we will explore this concept further and answer the question: if f is continuous on (−∞, ∞), what can you say about its graph?

    Theorem 1: If f is continuous on (−∞, ∞), then its graph is a connected set

    A function is continuous if given any two real numbers a and b, no matter how close together they are, we can always find a corresponding value c between them such that |f(a) – f(b)| < ε. This means that the graph of a continuous function will always be a connected set.

    Theorem 2: If f is continuous on (−∞, ∞), then its graph is a closed set

    Theorem 2: If f is continuous on (−∞, ∞), then its graph is a closed set.

    This theorem states that if a function is continuous over an infinite interval, then its graph will be a closed set. A closed set is defined as a set that contains all of its limit points. In other words, a closed set is a complete set – it contains everything it should contain, and nothing that it shouldn’t.

    Corollary: If f is continuous on (−∞, ∞) and F has an inverse function, then the graph of F is also a connected set

    If F is continuous on (−∞, ∞), and if F has an inverse function, then the graph of F is also a connected set.

    Conclusion

    In conclusion, when a function is continuous on an infinite domain such as (-∞, ∞), it means that its graph will be smooth and without any breaks or gaps. This can be seen by examining the definition of continuity: a function is continuous at a point iff its values approach each other from both sides of this point. The implication of this for the graph of F(-∞, ∞) is that there are no abrupt jumps in the curve which would indicate discontinuity; instead, the graph has one smooth line which connects all points across the domain.

  2. Paterson
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    2023-03-08T10:57:05+00:00
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    Are you curious about what the graph of a continuous function looks like?

    If the function is continuous on the interval of negative infinity to positive infinity (or, simply, (−∞, ∞), then you can say that its graph is smooth and continuous.

    A continuous function is one that has no breaks or gaps in its graph. This means that if we were to zoom in on any given part of the graph, that part would look just as smooth and continuous as the entire graph.

    Continuous functions are also known as “smooth” functions because they have no abrupt changes or jumps in their graph. ️ This means that the rate of change of the function is continuous and it is always increasing or decreasing at a steadily increasing or decreasing rate.

    The graph of a continuous function on the interval of (−∞, ∞) would also look like a line with no sharp corners or breaks. This means that the graph is connected, meaning that any two points on the graph can be connected by a line without having to go around a sharp corner or jump.

    Finally, a continuous function on the interval of (−∞, ∞) would also have a continuous derivative. This means that if you were to take the derivative of the function at any point on the graph, that derivative would also be continuous.

    In short, if a function is continuous on the interval of (−∞, ∞), you can say that its graph is smooth, connected, and has a continuous derivative.

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