1549901369-Elements_of_Real_Analysis__Denlinger_

5.1 Continuity of a Function at a Point 227

If Ix - 21 < 1, then -1 < x - 2 < 1, so 1 < x < 3, and so

3 < 3x < 9

7 < 3x + 4 < 13

l3x + 41<13.

c

Thus, we want to make sure that t5 ::::; 1 and t5 ::::;

13

.

(b) Proof: Let c: > 0. Choose t5 = min { 1,

1

c:

3

}. Then

c

Ix - 2[ < t5 ::::} [x - 2[ < 1 and [x - 2[ <

13

c

::::} -1 < x - 2 < 1 and Ix - 2[ < -

c 13

::::} 1 < x < 3 and [x - 21 <

13

c

::::} 3 < 3x < 9 and [x - 2[ < -

1 3 c

::::} 7 < 3x + 4 < 13 and [x - 2[ < -

c 13

::::} l3x + 41 < 13 and [x - 21 < -

c 13

::::} [3x + 4[/x - 2[ < 13 ·

13

::::} [ 3x^2 - 2x - 8 [ < c:

::::} /(3x^2 - 2x - 1) - 7[ < c:.

Therefore, the function f(x) = 3x^2 - 2x - 1 is continuous at x 0 = 2. D

As we have already suggested, sequences play a significant role in virtually
all areas of real analysis. The topic of continuity is no exception.

Theorem 5.1.3 (Sequential Criterion for Continuity off at x 0 )
A function f : D(f) --+ JR. is continuous at a point xo E D(f) iff \:/ sequences
{xn} in D(f) 3 Xn--+ xo, f(xn)--+ f(xo).

Proof. Exercise 6. (Compare with Theorem 4.1.9.) •

Corollary 5.1.4 (Sequential Criterion for Discontinuity off at x 0 )
A function f : D(f) --+ JR., is discontinuous at a point x 0 E D(f) iff 3
sequence {xn} in D(f) 3 Xn--+ xo, but {f(xn)} does not converge to f(x 0 ).

Example 5.1.5 The signum function, sgn(x) = { l~I if x ~ O}, is discon-
0 ifx=O
tinuous at x = 0.