- #1
0kelvin
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Take for ex f(x,y) = x/y. Domain is all (x,y) except for y = 0. It's continuous everywhere except for y = 0. Is this always the case? The function is continuous everywhere in its domain?
No. E.g. take ##f(x) = \begin{cases} 1 & \text{ if } x= 0\\ 0 & \text{ if } x\neq 0 \end{cases}##. It's defined for all real numbers, so its domain is ##\mathbb{R}## whereas it is not continuous at ##x=0##. Of course you can easily find more complicated examples.0kelvin said:Take for ex f(x,y) = x/y. Domain is all (x,y) except for y = 0. It's continuous everywhere except for y = 0. Is this always the case? The function is continuous everywhere in its domain?
Continuity in a function means that the function has no breaks or jumps in its graph. This means that as x approaches a certain value, the y-values of the function approach a specific value as well.
The largest subset in which a function is continuous can be determined by looking at the domain of the function. The subset will include all values within the domain that do not cause the function to be discontinuous.
Yes, it is possible for a function to be continuous on its entire domain. This means that there are no breaks or jumps in the graph of the function and it is continuous at every point within its domain.
Polynomial, exponential, and trigonometric functions are typically continuous on their entire domain. These types of functions do not have any breaks or jumps in their graphs and are continuous at every point within their domains.
If a function is not continuous on its entire domain, it means that there are values within the domain that cause the function to be discontinuous. This can result in breaks or jumps in the graph of the function, and the function may not have a limit at those points.