
In
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...
, the intermediate value theorem states that if
is a
continuous function whose
domain contains the
interval , then it takes on any given value between
and
at some point within the interval.
This has two important
corollaries:
# If a continuous function has values of opposite sign inside an interval, then it has a
root
In vascular plants, the roots are the plant organ, organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often bel ...
in that interval (Bolzano's theorem).
# The
image
An image or picture is a visual representation. An image can be Two-dimensional space, two-dimensional, such as a drawing, painting, or photograph, or Three-dimensional space, three-dimensional, such as a carving or sculpture. Images may be di ...
of a continuous function over an interval is itself an interval.
Motivation
This captures an intuitive property of continuous functions over the
real number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s: given ''
'' continuous on