Continuous functions: discontinuities and the zero, intermediate value and extreme value theorems
Definition of continuity, types of discontinuity, the zero, intermediate value and extreme value theorems, continuity of the inverse and uniform continuity.
Intuitively a function is continuous if its graph can be drawn without lifting the pen. The rigorous definition uses limits, and three theorems follow from it that support a good part of calculus: zeros, intermediate values, maxima and minima.
Definition
is continuous at 
(a point of the domain) if 
. Three things are needed: 
must exist, the limit must exist, and the two must agree. is continuous on a set if it is continuous at each of its points.
All elementary functions are continuous on their domain: polynomials, rational functions, roots, exponentials, logarithms, trigonometric functions and their inverses. Sums, products, quotients (where the denominator does not vanish) and compositions of continuous functions are continuous.
Points of discontinuity
| Type | What happens | Example |
|---|---|---|
| Removable | the limit exists and is finite but differs from , or is not defined there |  at 0 |
| Jump | left and right limits are finite but different | the step,  at 0 |
| Essential | at least one one-sided limit is infinite or does not exist |  at 0,  at 0 |
A removable discontinuity is “repaired” by redefining the function at the point: the sinc function equals for 
and 1 at 0, and it is continuous everywhere.
Zero theorem and intermediate value theorem
Intermediate zero theorem. If is continuous on ![[a, b]](https://www.electronics-engineering.com/wp-content/ql-cache/quicklatex.com-69b0b734c4059de07bf0a2619ad6df3a_l3.png "Rendered by QuickLaTeX.com")
and 
, there is 
with 
. The proof by bisection, with a numerical example, is in the in-depth article.
Intermediate value theorem. A function continuous on takes every value between 
and 
. It follows from the previous one applied to 
. Consequence: the image of an interval under a continuous function is again an interval.
Extreme value theorem (Weierstrass)
A function continuous on a closed and bounded interval has an absolute maximum and minimum. All hypotheses are needed:
- on
![(0, 1]](https://www.electronics-engineering.com/wp-content/ql-cache/quicklatex.com-1bb88c543f46a4661780397b742fee9a_l3.png "Rendered by QuickLaTeX.com")
, open on the left, has no maximum; - on
, unbounded, 
has no maximum; - a discontinuous function on
![[0, 1]](https://www.electronics-engineering.com/wp-content/ql-cache/quicklatex.com-61986df0a4507c5fa0a238a1e7db8406_l3.png "Rendered by QuickLaTeX.com")
equal to for 
and to 0 at 1 has no maximum: it gets close to 1 without reaching it.
The proof uses the Bolzano-Weierstrass theorem: take a sequence of points where approaches its supremum, extract a convergent subsequence and use continuity.
Continuity and invertibility
A continuous injective function on an interval is strictly monotone, and its inverse is continuous too. This is what guarantees the continuity of 
, 
, ![\sqrt[n]{x}](https://www.electronics-engineering.com/wp-content/ql-cache/quicklatex.com-96b5b1f91987d68a4430cbba410d0090_l3.png "Rendered by QuickLaTeX.com")
from that of 
, 
, 
.
Uniform continuity (outline)
In the definition of continuity, 
may depend on the point. If a single works for all points, the function is uniformly continuous. on 
is continuous but not uniformly: near 0 it gets steeper and steeper. The Heine-Cantor theorem ensures that on a closed bounded interval continuity and uniform continuity coincide. It is used to prove that continuous functions are integrable.
A typical exercise
For which value of 
is the function equal to 
for and to for 
continuous? The limit at 0 is 
(standard limit 
with 
), so 
.
In electronics continuity has a precise physical meaning: the voltage across a capacitor and the current in an inductor cannot jump, because that would require infinite power. It is the rule used to find the initial conditions of transients.
