Below you will find example sentences with "continuous function". The examples show how this phrase is used in real sentences and which words often surround it.
Continuous Function in a sentence
About this phrase
- Belongs to the word: continuous
Example types with continuous function
Below, the examples are grouped by length and sentence type:
A lower semi-continuous function. (5 words)
Every continuous function is sequentially continuous. (6 words)
Examples An upper semi-continuous function. (6 words)
Then ρ will be the finest completely regular topology on X which is coarser than τ. This construction is universal in the sense that any continuous function : to a completely regular space Y will be continuous on (X, ρ). (39 words)
Intuitively we expect that when one reduces a continuous function to a discrete sequence and interpolates back to a continuous function, the fidelity of the result depends on the density (or sample rate ) of the original samples. (37 words)
History The first published definition of uniform continuity was by Heine in 1870, and in 1872 he published a proof that a continuous function on an open interval need not be uniformly continuous. (33 words)
Example sentences (20)
Intuitively we expect that when one reduces a continuous function to a discrete sequence and interpolates back to a continuous function, the fidelity of the result depends on the density (or sample rate ) of the original samples.
The image of every compact set under a continuous function is compact, and the image of every connected set under a continuous function is connected.
Any absolutely continuous function is uniformly continuous.
Central to the theory was the concept of uniform continuity and the theorem stating that every continuous function on a closed interval is uniformly continuous.
Every continuous function is sequentially continuous.
History The first published definition of uniform continuity was by Heine in 1870, and in 1872 he published a proof that a continuous function on an open interval need not be uniformly continuous.
Informally, this example can be seen as taking the usual uniformity and distorting it through the action of a continuous yet non-uniformly continuous function.
More generally, every Hölder continuous function is uniformly continuous.
Then ρ will be the finest completely regular topology on X which is coarser than τ. This construction is universal in the sense that any continuous function : to a completely regular space Y will be continuous on (X, ρ).
Boundedness Consider the function : which is a continuous function from R to itself.
Closedness Consider the function : which is a continuous function from the open interval (−1,1) to itself.
Consider the following function, defined in polar coordinates: : which is a continuous function from the unit circle to itself.
A continuous function in L 1 might blow up near 0 but must decay sufficiently fast toward infinity.
A lower semi-continuous function.
Among other things, he showed that every piecewise continuous function is integrable.
A typical application of the extendability of a uniform continuous function is the proof of the inverse Fourier transformation formula.
A uniformly continuous function is defined as one where inverse images of entourages are again entourages, or equivalently, one where the inverse images of uniform covers are again uniform covers.
Condition 2. is satisfied because players maximize expected payoffs which is a continuous function over a compact set.
Every continuous function : is integrable (for example in the sense of the Riemann integral ).
Examples An upper semi-continuous function.