Question

What does it mean when a function is continuous from the right?

Answer 1

By definition, a function is continuous at a point a in its domain if
\[\lim_{x \rightarrow a} \ f(x) = f(a)\]

To be continuous from the right means that the limit from the right equals the value of the function
\[\lim_{x \rightarrow a+} \ f(x) = f(a)\]

It's clear any function continuous at a is also continuous from the right at a.

An example of a function that is continuous from the right at a, but not continuous at a, is the following:
\[f(x) = \left\{\begin{matrix} 1 & x \geq a \\ 0 & x < a \end{matrix}\right.\]
as f(a) = 1 and the limit at a from the right is also 1 but the limit at a itself doesn't exist.

Answer 2

There is an analogous definition for continuous from the left.

Answer 3

Thank you.