We've observed that the nice functions that we usually work with have the property that ![[Graphics:LimitsTutorialgr137.gif]](LimitsTutorialgr137.gif)
f[x] = f[a] for all points in their domain. In B.6) we gave a name to these functions that have really nice limits---we said that they were continuous functions. Intuitively f[x] is continuous at x = a, if it has no difficuties to overcome at x = a. In other words, the graph has no hole, jump, or wild wiggles at x = a.
The formal definitions of continuous at a point, x = a, continous on an open interval (a, b), and continuous on a closed interval [a, b] are all given in B.6.
In this section, we give mathematical proofs (i.e. &egr;--&dgr; proofs) that linear functions (including constant functions) and power functions are continuous at every point in their domain. Proofs that e^x, and Log[x] are continous are similar to the proof for power functions and are left as exercises for the reader. We also make mention of some important properties of continous functions.
Use the definitions of limit and continuity to prove that the linear function, f[x] = m x + b, is continous at every real number.
Let r not equal to 0 be given. Use the definitions of limit and continuity to prove that the power function, f[x] = x^r, is continous every where it is defined.
Because of the limit theorems, it's easy to show that if f[x] and g[x] are continuous at x = c, then so are the functions:
f[x] ± g[x], k f[x], and f[x] g[x].
If g[c] is not 0, then it's also easy to show that f[x]/g[x] is continous at x = c.
Compositions of functions are a bit more difficult. Suppose that g[x] is continuous at x = c, and that f[x] is continuous at g[c]. Is the composition, f[g[x]] continuous at x = c? Why?
Use the definitions of limit and continuity to show that if f[x] and g[x] are continuous at all real numbers then so is the composition function, f[g[x]].