Math 320, December 09, 2018 16 Continuous functions Definition 16.1. Let f : D ! R and let c 2 D. We say that f is continuous at c, if for every > 0 there exists δ>0 such that jf(x)−f(c)j<, whenever jx−cj<δ and x2D. If f is continuous at every point of a subset S ⊂D, then f is said to be continuous on S. If f is continuous on its domain D, then f is said to be continuous. Notice that in the above definition, unlike the definition of limits of functions, c is not required to be a limit point of D. It is clear from the definition that if c is an isolated point of the domain D, then f must be continuous at c. The following statement gives the interpretation of continuity in terms of neighborhoods and sequences. Theorem 16.2. Let f :D!R and let c2D. Then the following three conditions are equivalent. (a) f is continuous at c (b) If (xn) is any sequence in D such that xn !c, then limf(xn)=f(c) (c) For every neighborhood V of f(c) there exists a neighborhood U of c, such that f(U \D)⊂V . If c is a limit point of D, then the above three statements are equivalent to (d) f has a limit at c and limf(x)=f(c). x!c From the sequential interpretation, one gets the following criterion for discontinuity. Theorem 16.3. Let f :D !R and c2D. Then f is discontinuous at c iff there exists a sequence (xn)2D, such that (xn) converges to c, but the sequence f(xn) does not converge to f(c). Since operations for functions can be extended to operations for limits, we have the following result for continuity of sums, products and ratios of functions. Theorem 16.4. Let f :D!R, g:D!R and c2D. Suppose k2R, f and g are continuous at c. Then (f +g), (kf), (fg) are continuous at c, and (f=g) is continuous at c, whenever g(c)=06 . Finally, we also have the expected continuity of compositions. Theorem 16.5. Let f :D !R and g :E !R be two functions, such that f(D)⊂E. If f is continuous at c and g is continuous at f(c), then g◦f :D!R is continuous at c. Another equivalent definition of continuity which directly generalizes to any topological (hence metric) spaces is given by the following. Theorem 16.6. A function f :D!R is continuous on D iff for every open set G in R there exists an open set H in R, such that H\D=f−1(G). If a function has the entire real line as its domain, then the above theorem implies. Corollary 16.7. A function f :R!R is continuous iff f−1(G) is open in R whenever G is open in R..