CONTINUITY OF SCALAR VALUED MULTIVARIABLE FUNCTIONS

$f:D\rightarrow ℝ$ is a scalar valued multivariable function, where $D \in ℝ^d$. $f(x)$ is continuous at $v \in D$ if the following conditions are true

• Limit of $f$ at $v$ must exist

• The function should be defined at $v$

• $\lim\limits _ {x \rightarrow v} f(x)=f(v)$

In simple words, the limit of $f$ must exists at $v$, the function should be defined at $v$ and the value of the function at $v$ should be equal to the limit of the function at $v$.

If a function is continuous at a point, then the limit at that point can be found by evaluating the function at that point.

CONTINUOUS FUNCTION

A function is said to be continuous, if the function is continuous at all the points in its domain.