Continuous Function

A continuous function is a Function $f:X \rightarrow Y$ where the pre-image of every Open Set in $Y$ is Open in $X$. A function $f(x)$ in a single variable $x$ is said to be continuous at point $x_0$ if

1. $f(x_0)$ is defined, so that $x_0$ is in the Domain of $f$.

2. $\lim_{x\to x_0} f(x)$ exists for $x$ in the Domain of $f$.

3. $\lim_{x\to x_0} f(x) = f(x_0)$,

where lim denotes a Limit. If $f$ is Differentiable at point $x_0$, then it is also continuous at $x_0$. If $f$ and $g$ are continuous at $x_0$, then

1. $f+g$ is continuous at $x_0$.

2. $f-g$ is continuous at $x_0$.

3. $f\times g$ is continuous at $x_0$.

4. $f\div g$ is continuous at $x_0$ if $g(x_0) \not = 0$ and is discontinuous at $x_0$ if $g(x_0) = 0$.

5. $f\circ g$ is continuous at $x_0$, where $f\circ g$ denotes $f(g(x))$, the Composition of the functions $f$ and $g$.

See also Critical Point, Differentiable, Limit, Neighborhood, Stationary Point