**Definition:**

(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;

**NOTE:**

- 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**.

**Definition**:

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)*.

__For e.g;__

(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**.

**Theorem**:

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.

**NOTE:**

- 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**:

(i)

(ii)

(iii)

(iv) if

(v)

** 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

= 9

(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 .

UCT MAM1000 lecture notes subject links – MathemafricaApril 2, 2016 at 1:02 pm[…] Continuity – part 2 […]

Desmond DyasiApril 3, 2016 at 2:58 pmis it possible to get an interval I= (a,b) where a function is continuous on every point on the graph except a and b?

Azhar RohimanApril 4, 2016 at 8:00 amYes, it is. The interval will be [a+1, b-1].

TseboMarch 27, 2019 at 2:03 pmLove this ,thank you