Continuity for Real functions
We now introduce the second important idea in Real analysis.
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.
- A function f : R→ R is said to be continuous at a point p ∈ R if whenever (an) is a real sequence converging to p, the sequence (f (an)) converges to f (p).
- A function f defined on a subset D of R is said to be continuous if it is continuous at every point p ∈ D.
- 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.
- 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.
- 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.
- 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 .
- This follows directly from the corresponding arithmetic properties of sequences.
For example: to prove that f + g is continuous at p ∈ R
Suppose (xn)→ p. We are told that (f (xn))→ f (p) and (g(xn))→ g(p) and we must prove that (f + g)(xn))→ (f + g)(p).
But the LHS of this expression is f (xn) + g(xn) and the RHS is f (p) + g(p) and so the result follows from the arithmetic properties of sequences.
- The composite of continuous functions is continuous.
- 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 (xi)→ p. Then (f (xi))→ f (p) and then (g(f (xi)))→ g(f (p)) which is what we need.
- 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 sin2(x + 5), exp(-x2), √(1 + x4), ... are continuous since they are made by composing continuous functions.
JOC September 2001