Continuity of a Function at an input value

Value of a function

» Value of f(x)f(x)

→ Evaluated at input f(x)∣x=af(x)∣x=a or f(a)f(a)

→ Left-hand-limit limx→a-f(x)limx→a−f(x)

→ Right-hand-limit limx→a+f(x)limx→a+f(x)

» A function f(x)f(x) at x=a is

→ **continuous**: if f(a) = LHL = RHL

→ **defined by value**: if f(a) is a real number

→ **defined by limit**: if f(a)=00 and LHL = RHL

→ **not defined**: if LHL ≠ RHL and f(a)∉ℝ

Limit of a Function

» If both left-hand-limit and right-hand-limit are equal, it is together referred as "limit of the function"

another motivation

So far, the motivation to examine limits of a function was * to evaluate the function at a input value of the argument variable where the function evaluates to 'indeterminate value'*.

In this topic, another motivation to examine limits is explained.

Consider f(x)=1x-1 at x=1

by directly substituting x=1

f(1)=11-1=+∞

limx→1+f(x)

=11+δ-1

=1δ

=10=+∞

limx→1-f(x)

=11-δ-1

=1-δ

=-10=-∞.

For the function f(x)=1x-1,

•
f(a)=∞

•
limx→1+f(x)=∞

•
limx→1-f(x)=-∞

The plot of the function is given in the figure.

That is, for a value less than x=1, the function is -∞. And at x=1, the function becomes ∞.

*The function is not continuous.*

continuous

A function f(x) at a given input value x=a is continuous if all the three are equal

f(x)∣x=a

=limx→a-f(x)

=limx→a+f(x)

The word 'continuous' means: unbroken and continue from one side to another without pause in between.

A function is *continuous* at an input value, if the following three are equal

• function evaluated at the input

• left-hand limit of the function at that input value and

• right-hand limit of the function at that input value.

example

Given function f(x)=2x2, is it continuous at x=0?

The answer is 'Yes, Continuous'. Evaluate the three values of the function and they are equal.

summary

**Continuity of a Function: **A function f(x) is continuous at x=a if all the following three have a *defined value* and are *equal*

• Evaluated at the input value f(x)∣x=a

• left-hand-limit limx→a-f(x)

• right-hand-limit limx-a+f(x)

limit of a function

Given that function f(x) evaluates to indeterminate value at x=a. To evaluate the expected value of f(x)∣x=a, we examine ;

• Left-hand-limit limx→a-f(x)

• Right-hand-limit limx→a+f(x)

If these two limits are equal then the result is referred as "*limit of the function at the input value*" limx→af(x)

The significance of this is that, most functions have both right-hand-limit and left-hand-limit equal.

summary

** Limit of a function: ** Given function f(x) and that f(x)∣x=a=00.

If limx→a+f(x)=limx→a-f(x),

then the common value is referred as limit of the function limx→af(x).

discontinuous

If a function f(x) is discontinuous at x=a, then what is limx→af(x)?

The answer is 'cannot be computed'. It is given that the function is discontinuous at x=a, and that implies left-hand-limit and right-hand-limits are not equal. In that case, limit of the function cannot be computed without specifying left or right.

summary

**Continuity of a Function: **A function f(x) is continuous at x=a if all the following three have a *defined value* and are *equal*

• Evaluated at the input value f(x)∣x=a

• left-hand-limit limx→a-f(x)

• right-hand-limit limx-a+f(x)

** Limit of a function: ** Given function f(x) and that f(x)∣x=a=00.

If limx→a+f(x)=limx→a-f(x),

then the common value is referred as limit of the function limx→af(x)

Outline

The outline of material to learn "limits (calculus)" is as follows.

Note : * click here for detailed outline of Limits(Calculus).*

→ __Indeterminate and Undefined__

→ __Indeterminate value in Functions__

→ __Expected Value__

→ __Continuity__

→ __Definition by Limits__

→ __Geometrical Explanation for Limits__

→ __Limit with Numerator and Denominator__

→ __Limits of Ratios - Examples__

→ __L'hospital Rule__

→ __Examining a function__

→ __Algebra of Limits__

→ __Limit of a Polynomial__

→ __Limit of Ratio of Zeros__

→ __Limit of ratio of infinities__

→ __limit of Binomial__

→ __Limit of Non-algebraic Functions__