Conceptual question concerning functions

In summary, if a function is not defined at a point, then that point is not in its domain. However, a function can still be continuous at each point of its domain even if its domain consists of disjoint subsets of a larger set.
  • #1
Duderonimous
63
1

Homework Statement



If I have a function that is not defined at a point in its domain is this the same as saying it is discontinuous?

Homework Equations





The Attempt at a Solution

 
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  • #2
Duderonimous said:

Homework Statement



If I have a function that is not defined at a point in its domain is this the same as saying it is discontinuous?

Homework Equations


The Attempt at a Solution


If it's not defined at some point, then that point is not in the domain of that function.

If the function is not defined at some point, then it is discontinuous at that point.
 
  • #3
Duderonimous said:
If I have a function that is not defined at a point in its domain is this the same as saying it is discontinuous?

No. If a function is not defined at a point, then that point is not in its domain. But a function can be continuous at each point of its domain notwithstanding that its domain consists of disjoint subsets of some larger set. For example, the following function is continuous everywhere in its domain:
[tex]
f : \mathbb{R} \setminus \{0\} \to \mathbb{R} : x \mapsto \left\{\begin{array}{r@{\qquad}l}
0 & x < 0 \\
1 & x > 0
\end{array}\right.
[/tex]
On the other hand, there is no [itex]a \in \mathbb{R}[/itex] such that the following function is continuous at 0:
[tex]
g : \mathbb{R} \to \mathbb{R} : x \mapsto \left\{\begin{array}{r@{\qquad}l}
f(x) & x \neq 0 \\
a & x = 0
\end{array}\right.
[/tex]
 

What is a function?

A function is a mathematical concept that represents a relationship between two sets of values, known as the input and output. It takes an input value and produces a corresponding output value based on a specific rule or set of rules.

What is the difference between a one-to-one function and a many-to-one function?

A one-to-one function is a type of function where each input value has exactly one corresponding output value. In other words, there are no repeated output values. On the other hand, a many-to-one function is a type of function where multiple input values can have the same output value. This means that there can be repeated output values.

What is the domain and range of a function?

The domain of a function is the set of all possible input values for which the function is defined. The range, on the other hand, is the set of all possible output values that the function can produce. In other words, the domain is the set of all x-values and the range is the set of all y-values of a function.

How do you determine if a graph represents a function?

A graph represents a function if it passes the vertical line test. This means that if a vertical line is drawn anywhere on the graph, it should only intersect the graph at one point. If the vertical line intersects the graph at more than one point, then the graph does not represent a function.

What is the difference between a linear and a nonlinear function?

A linear function is a type of function where the output value changes at a constant rate as the input value changes. This results in a straight line when graphed. A nonlinear function, on the other hand, is a type of function where the output value changes at a non-constant rate as the input value changes. This results in a curve when graphed.

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