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Mathematical Expression Editor
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The limit of a continuous function at a point is equal to the value of the function at that point.
Video Lecture
Explanation
Limits are simple to compute when they can be found by plugging the value into the function. That is, when We call this property
continuity.
A function is continuous at a point if
(a)
exists.
(b)
exists (i.e. is in the domain of ).
(c)
.
Often in literature you may see the alternative definition that is continuous at if . This is actually the same thing, click the right
blue arrow to the right if you want to know more!
This more compact form of “a function is continuous at a point ” is really making three statements:
(a)
is defined. That is, is in the domain of .
(b)
exists.
(c)
.
Notice that can’t fail to exist and still equal the limit of something else. In essence, the equality sign here requires both sides to
exist before equality can be evaluated, thus writing just the last statement implies the first two.
Actual mathematicians love brevity and succinctness in their definitions; so they will commonly use the alternate definition as it is
shorter and equivalent. But this brevity and succinctness that mathematicians value can often make it harder on students first
learning a concept.
1 : Consider the graph of the function
Which of the following are true?
Find the discontinuities (the points where a function is not continuous) for the function described below:
To start, is not even defined at , hence cannot be continuous at .
Next, from the plot above we see that does not exist because Since does not exist, cannot be continuous at
.
We also see that while . Hence , and so is not continuous at .
Building from the definition of continuity at a point, we can now define what it means for a function to be continuous on an open
interval.
A function is continuous on an open interval if for all in .
Loosely speaking, a function is continuous on an open interval if you can draw the function on that interval without any breaks in
the graph. This is often referred to as being able to draw the graph “without picking up your pencil.”
Continuity of Parent Functions The following functions are continuous on their natural domains, for a real number and a positive
real number:
Constant function
Identity function
Power function
Exponential function
Logarithmic function
In essence, we are saying that the functions listed above are continuous wherever they are defined.
2 : Compute:
The function is of the form for a positive real number . Therefore, is continuous for all positive real values of .
In particular, is continuous at . Since is continuous at , we know that . That is, .
3 : Compute:
The function is a constant. Therefore, is continuous for all real values of . In particular, is continuous at .
Since is continuous at , we know that . That is, .
4 : Compute:
The function is continuous for all real values of . In particular, is continuous at . Since is continuous at , we
know that . That is, .
Start typing the name of a mathematical function to automatically insert it.
(For example, "sqrt" for root, "mat" for matrix, or "defi" for definite integral.)
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Start typing the name of a mathematical function to automatically insert it.
(For example, "sqrt" for root, "mat" for matrix, or "defi" for definite integral.)