Conceptual question concerning functions

[Duderonimous]

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

 

[Mark44]

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.

 

[pasmith]

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:
$$f : \mathbb{R} \setminus \{0\} \to \mathbb{R} : x \mapsto \left\{\begin{array}{r@{\qquad}l}<br /> 0 & x < 0 \\<br /> 1 & x > 0<br /> \end{array}\right.$$
On the other hand, there is no $a \in \mathbb{R}$ such that the following function is continuous at 0:
$$g : \mathbb{R} \to \mathbb{R} : x \mapsto \left\{\begin{array}{r@{\qquad}l}<br /> f(x) & x \neq 0 \\<br /> a & x = 0<br /> \end{array}\right.$$