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- We now introduce the second important idea in Real analysis.
- A function
*f*:**R**→**R**is said to be**continuous at a point**if whenever (*p*∈ R*a*_{n}) is a real sequence converging to*p*, the sequence (*f*(*a*_{n})) converges to*f*(*p*). **Definition**- A function
*f*defined on a subset*D*of**R**is said to be**continuous**if it is continuous at every point*p*∈*D*. **Example**- In the discontinuous function above take a sequence of reals converging to
*c*from below. (That is, all the terms are <*c*.) Then the image of these gives a sequence which does not converge to*f*(*c*).We also have the following.

**Definition**- A real valued function
*f*defined on a subset*S*of**R**is said to be**continuous**if it is continuous at all points of*S*. - It will be easier to give (a lot of) examples of continuous functions after we have proved the following two results.
**Definition**

- If
*f*and*g*are functions from**R**to**R**, we define the function*f*+*g*by (*f*+*g*)(*x*) =*f*(*x*) +*g*(*x*) for all*x*in**R**.

Similarly we may define the difference, product and quotient of functions. **Theorem**

*If f and g are continuous a point p of***R***, then so are f*+*g, f - g, f.g and*(*provided g*(*p*) ≠ 0)^{f}/_{}g .**Proof**

- This follows directly from the corresponding arithmetic properties of sequences.

For example: to prove that*f*+*g*is continuous at*p*∈**R**

Suppose (*x*_{n})→*p*. We are told that (*f*(*x*_{n}))→*f*(*p*) and (*g*(*x*_{n}))→*g*(*p*) and we must prove that (*f*+*g*)(*x*_{n}))→ (*f*+*g*)(*p*).

But the LHS of this expression is*f*(*x*_{n}) +*g*(*x*_{n}) and the RHS is*f*(*p*) +*g*(*p*) and so the result follows from the arithmetic properties of sequences.

**Theorem**

*The composite of continuous functions is continuous.***Proof**

- Suppose
*f*:**R**→**R**and*g*:**R**→**R**. Then the composition*g**f*is defined by*g**f*(*x*) =*g*(*f*(*x*)).

We assume that*f*is continuous at*p*and that*g*is continuous at*f*(*p*). So suppose that (*x*_{i})→*p*. Then (*f*(*x*_{i}))→*f*(*p*) and then (*g*(*f*(*x*_{i})))→*g*(*f*(*p*)) which is what we need.

**Examples**- Clearly the identity function which
*x*↦*x*is continuous.

Hence, using the above, any polynomial function is continuous and hence any*rational function*(a ratio of polynomial functions) is continuous at any point where the denominator is non-zero. - We will see later that functions like √, sin, cos, exp, log, ... are continuous. It follows that , for example sin
^{2}(*x*+ 5), exp(-*x*^{2}), √(1 +*x*^{4}), ... are continuous since they are made by composing continuous functions.

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- Clearly the identity function which

Continuity can be defined in several different ways which make rigorous the idea that a continuous function has a graph with no breaks in it or equivalently that "close points" are mapped to "close points".

For example, is the graph of a continuous function on the interval (*a*, *b*)

while is the graph of a function with a discontinuity at *c*.

To understand this, observe that some points close to *c* (arbitrarily close to the left) are mapped to points which are not close to *f* (*c*).

We will give a definition in terms of convergence of sequences and show later how it can be reformulated in terms of the above description.

**Definition**