(i) A function is said to be continuous from the right at a if
We can see that, as the function approaches a certain x-value from the right, is defined and
And as the function approaches a certain x-value from the left, is not defined, i.e;
Therefore, we say that the function is continuous from the right at this point, but is discontinuous from the left.
(ii) A function is said to be continuous from the left at a if
Here, it is clear from the graph that the function is continuous from the left as x approaches 3. This is because the function is defined at x = 3 and,
However, from the right,
So, we can say that the function is continuous from the left, but discontinuous from the right, at the point x = 3.
We may also have a function where it is neither continuous at a point from the left or from the right but is defined elsewhere. In that case, the graph may look like this;
- is continuous at a if and only if is continuous at a from the left and from the right. This can be written as:
If you choose any x-value, you will notice that the function is continuous everywhere. This is because
for any x.
A function is said to be continuous on an interval if it is continuous at each and every point in the interval. Continuity at an endpoint, if it exists, means is continuous from the right (for the left endpoint) or continuous from the left (for the right endpoint).
(i) If , is continuous from the right at a, and continuous at every point in the interval.
(ii) If , is continuous from the left at b, and continuous at every other point in the interval.
(iii) If , is continuous from the right at a and continuous from the left at b.
The following functions are continuous at every point of their domain.
(a) Polynomial functions.
(b) Rational functions. (The discontinuities of a rational function occur at the zeros of it’s denominator.)
(c) Exponential functions.
(d) Trigonometric functions.
(e) Logarithmic functions.
(f) Root functions.
- Inverse function of a continuous function is continuous.
Continuity of the algebraic combinations of functions.
If and are both continuous at x = a and c is any constant, then each of the following functions is also continuous at a:
Continuity of composite functions.
If is continuous at a, and is continuous at , then the composite function is also continuous at a.
Let’s look at an example.
Let , & a = 2.
We know that
exists here and is equal to 3. Now, we also know that is continuous at . And, when evaluating the limit of at , we have
(Remember that 3)
The statement says that if those two conditions above hold, then is also continuous at a. This means
Another way to put this is:
If is continuous at a, we evaluate first as it is the starting function. This gives us a value back, and if the value given back is in the domain of (or, if is continuous at ), we evaluate the function at .