Continuous function
A mathematical function is called continuous if, roughly said, a small change in the input only causes a small change in the output. If this is not the case, the function is discontinuous. Functions defined on the real numbers, with one input and one output variable, will show as an uninterrupted line (or curve). They can be drawn without lifting the pen off of the page. The definition given above was made by Augustin-Louis Cauchy.[1]
Karl Weierstraß gave another definition of continuity: Imagine a function f, defined on the real numbers. At the point [math]\displaystyle{ x_0 }[/math] the function will have the value [math]\displaystyle{ f(x_0) }[/math]. If the function is continuous at [math]\displaystyle{ x_0 }[/math], then for every value of [math]\displaystyle{ \varepsilon\gt 0 }[/math] no matter how small it is, there is a value of [math]\displaystyle{ \delta \gt 0 }[/math], so that [math]\displaystyle{ |x - x_0| \lt \delta }[/math], means that [math]\displaystyle{ |f(x) - f(x_0)| \lt \varepsilon }[/math]. We can put this another way, given a point close to [math]\displaystyle{ x_0 }[/math] (called x), the absolute value of the difference between the two values of the function can be made increasingly small, if the point x is close enough to [math]\displaystyle{ x_0 }[/math].
There are also special forms of continuous, such as Lipschitz-continuous. A function is Lipschitz-continuous if there is a [math]\displaystyle{ L }[/math] with [math]\displaystyle{ |f(x) - f(y)| \le L|x - y| }[/math] for all x,y ∈ (a,b).
A basic way to know if a function is continuous is to use a pencil or your finger. Then, start at the left of the function. Then, move your finger along the path of the function. If you ever need to lift your finger or pencil to keep following the function, then you know it is not continuous. This is because, by lifting your finger, you have "jumped" from one section of the function to another. That means you made a very small movement but the function changed very much. This is what the first sentence of this article is talking about.
Continuous Function Media
The function f(x)=\tfrac 1 x is continuous on its domain (\R\setminus \{0\}), but discontinuous (not-continuous or singularity) at x=0.Nevertheless, the Cauchy principal value can be defined. On the other hand, in complex analysis (\mathbb{C}, especially \widehat{\mathbb{C}}.), this point (x=0) is not regarded as "undefined" and it is called a singularity, because when thinking of x as a complex variable, this point is a pole of order one, and then Laurent series with at most finite principal part can be defined around the singular points.
The sequence exp(1/n) converges to exp(0) = 1
Illustration of the ε-δ-definition: at x = 2, any value δ ≤ 0.5 satisfies the condition of the definition for ε = 0.5.
The failure of a function to be continuous at a point is quantified by its oscillation.
The graph of a cubic function has no jumps or holes. The function is continuous.
The graph of a continuous rational function. The function is not defined for x = -2. The vertical and horizontal lines are asymptotes.
The sinc and the cos functions
Plot of the signum function. It shows that \lim_{n\to\infty} \sgn\left(\tfrac 1 n\right) \neq \sgn\left(\lim_{n\to\infty} \tfrac 1 n\right). Thus, the signum function is discontinuous at 0 (see section 2.1.3).
Point plot of Thomae's function on the interval (0,1). The topmost point in the middle shows f(1/2) = 1/2.
Uniform continuity animation
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Related pages
References
- ↑ Fischer, Helmut; Helmut Kaul (2007). Mathematik für Physiker Band 1: Grundkurs. Teubner Studienbücher Mathematik. Teubner. p. 165 ff. ISBN 978-3-8351-0165-4.
