maths > calculus-limits

Function defined at an input value

what you'll learn...

A function Defined by Limits

»  A function f(x)$f \left(x\right)$ at x=a$x = a$ is
→  continuous: if f(a)$f \left(a\right)$ = LHL = RHL
→  defined by value: if f(a)$f \left(a\right)$ is a real number
→  defined by limit: if f(a)=00$f \left(a\right) = \frac{0}{0}$ and LHL = RHL
→  not defined: if LHL $\ne$ RHL and $f \left(a\right) \notin \mathbb{R}$

function, defined at a input

A function $f \left(x\right)$ is defined at the input value $x = a$, when $f \left(a\right)$ evaluates to a defined value.

If $f \left(a\right)$ evaluates to $\infty$ or $\frac{0}{0}$, then the function need to be further examined.

If the left-hand-limit and right-hand-limit take a defined value and are equal, then the function is defined at that input value.

This is understood from

•  In an application scenario, the value is not exact and so the function will evaluate to a stable value given by limits.

•  In an abstract scenario, the expected value at that input value can be taken to be the value of the function.

A function is defined at an input value, if the function evaluates to a definite value OR if the left-hand-limit and right-hand-limits are equal.

The word 'defined' means: give exactly; state or describe without any ambiguity.

summary

Function is defined: Given a function $f \left(x\right)$, the function is defined at $x = a$
•  if $f \left(a\right)$ is a defined value. OR

•  if $f \left(a\right)$ is an indeterminate value $\frac{0}{0}$ or undefined large $\infty$, then ${\lim}_{x \to a +} f \left(x\right) = {\lim}_{x \to a -} f \left(x\right)$

defined vs continuous

Difference between Defined and Continuous :
A function is defined if it evaluates to a definite value OR when is the function evaluates to indeterminate value $\frac{0}{0}$, then left-hand-limit and right-hand-limit are equal.

A function is continuous only when it evaluates to a definite value at the input value AND also left-hand-limit and right-hand-limit are equal to the definite value at the input value.

•  For a function to be continuous at an input value, it has to be defined at that input value.

•  If a function is defined at an input value, it does not imply it is continuous. It may be or may not be continuous.

•  If a function is not defined at an input value, it implies it is not continuous.

If a function is not continuous at an input value, can the function be defined at that input value?

'Yes.'. Theoretically a function can have a value $f \left(a\right)$ different to the limits.

summary

Function is defined: Given a function $f \left(x\right)$, the function is defined at $x = a$
•  if $f \left(a\right)$ is a defined value. OR

•  if $f \left(a\right)$ is an indeterminate value $\frac{0}{0}$ or undefined large $\infty$, then ${\lim}_{x \to a +} f \left(x\right) = {\lim}_{x \to a -} f \left(x\right)$

A function $f \left(x\right)$ at $x = a$ is
→  continuous: if $f \left(a\right)$ = LHL = RHL
→  defined by value: if $f \left(a\right)$ is a real number
→  defined by limit: if $f \left(a\right) = \frac{0}{0}$ and LHL = RHL
→  not defined: if LHL $\ne$ RHL and $f \left(a\right) \notin \mathbb{R}$

Outline