Continuity of Functions
Study the continuity of the following functions:
The function is a polynomial of degree zero.
The domain of a polynomial function is 
.
Thus, the function is continuous at all points.
The function is a polynomial of degree zero.
The domain of a polynomial function is .
Thus, the function is continuous at all points.
The function is a polynomial of degree one.
The domain of a polynomial function is .
Thus, the function is continuous at all points.
The function is continuous at all points in its domain except at values that make the denominator zero.
But the denominator is always positive, therefore its domain is
The function is continuous at all points.
The function is continuous at all points in its domain except at values that make the denominator zero.
The function has two points of discontinuity at 
and 
. 
The function is continuous throughout ℝ except at values where the denominator equals zero. If we set this equal to zero and solve the equation, we will obtain the points of discontinuity.
; and solving the 2nd degree equation we also obtain:
y 
The function has three points of discontinuity at , and
The function is continuous throughout ℝ.
Jump = 
The function has an unavoidable jump discontinuity of 
at 
.
At 
there is a finite jump discontinuity.
Jump = 
The function has an unavoidable jump discontinuity of 
at .
Continuity at x=0
Study the continuity at the origin of the following functions:
is not defined
At there is an essential discontinuity.
The function is continuous at
At there is an essential discontinuity.
At there is an infinite jump discontinuity.
The function 
is bounded 
. Therefore it holds that: 
The limit is 
, since any number multiplied by zero gives zero.
The function is continuous throughout ℝ.
Prove Continuity Where Indicated
Given the function:
Prove that 
is not continuous at 
.
We resolve the indeterminate form by factoring the numerator and simplifying:
is not continuous at 
because:
Given the function:
Does there exist a continuous function that coincides with for all values 
?
If 
the function would be continuous, therefore the redefined function is:
Study the continuity of the function:
The function is continuous for 
. Let's study the continuity at 
.
The function is not continuous at , because it is not defined at , since it makes the denominator zero.
Study the continuity of the function:
The function is continuous throughout ℝ.
Study, in the interval (0,3), the continuity of the function:
There is only doubt about the continuity of the function at points and 
, where the form of the function changes.
Jump = 
At it has a jump discontinuity of 
.
Jump = 
At 
it has a jump discontinuity of .
Calculate Values to Guarantee Continuity
Calculate the value of 
so that the following function is continuous:
The function is not continuous when 
This has a solution only if is negative or zero.
Therefore, is continuous if is positive.
Calculate the value of so that the following function is continuous:
Calculate the value of so that the following function is continuous:
Calculate the value of so that the following function is continuous:
The following function is defined by:
and is continuous on 
. Find the value of that makes this statement true.
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