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Thought-Process to Discover Knowledge

Welcome to nubtrek.

Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

Read in the blogs more about the unique learning experience at nubtrek.
mathsLimit of a functionBasics: Limit of a function

### Continuity of a Function at an input value

Even when functions have a defined value at an input value, the function may be discontinuous. Limit of the function at that input value can be used to determine continuity.

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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)=1/color(coral)(x-1) at x=1

by directly substituting x=1
f(1) = 1/(1-1) = +oo

Consider f(x)=1/color(coral)(x-1).

lim_(x->1+)f(x)

quad quad = 1/color(coral)(1+delta-1)
quad quad = 1/color(coral)(delta)
quad quad = 1/0 = +oo

Consider f(x)=1/color(coral)(x-1)

lim_(x->1-)f(x)
quad quad = 1/color(coral)(1-delta-1)
quad quad = 1/color(coral)(-delta)
quad quad = -1/0 = -oo.

For the function f(x)=1/color(coral)(x-1),
•  f(a) = oo
•  lim_(x->1+)f(x) = oo
•  lim_(x->1-)f(x) = -oo

The plot of the function is given in the figure.

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

The function is not continuous.

A function f(x) at a given input value x=a is continuous if all the three are equal
f(x)|_(x=a)
quad quad = lim_(x->a-) f(x)
quad quad = lim_(x->a+) f(x)

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

familiarize with the terminology
continuous

A function is continuous at an input value, if the function evaluated at the input equals both left-hand limit and right-hand limit of the function at that input value.

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 lim_(x->a-) f(x)

•  right-hand-limit lim_(x-a+) f(x)

Solved Exercise Problem:

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

• Yes, Continuous.
• Yes, Continuous.
• No, Not continous

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

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