Question

#3

(1) What does the continuity of a function f , as x approaches a real number a, mean?
Give examples-and NONexamples (i.e., what it means for f to not have continuity at
a)-to illustrate this graphically. Be sure to touch on one-sided continuity, to0. Be
prepared to show/explain, graphically, what, for example,
f is continuous at x=-3
f is continuous only from the right at x=0
f is discontinuous at x=2
means.
(2) Now show how to go about proving/disproving continuity at a algebraically. That
is, suppose you are given a function f. You want to know if f is continuous at a,
without pictures. What's the procedure? Explain it out. Be sure to give an example of a
piecewise function, also. For example, suppose f is the function
V4x² +1 x<0
f (x)=
0.
x=0. Is/f continuous at 0? If so, why? If not, why not?
x* +1
x>0
(3)(Based on all this, suppose you have the function f given by
ta'x? +3
f(x) = { a+9
x = 2
.2
ax?
x > 2
You are given that f is continuous at x=2. Find all possible values for a.
(4) Suppose you have the function f given by
4
x* - 1
x<1
f(x)=
5x³ - 9x2 +7 x21
3
Is f continuous at 1? If so, why? If not, why not?
expand button
Transcribed Image Text:(1) What does the continuity of a function f , as x approaches a real number a, mean? Give examples-and NONexamples (i.e., what it means for f to not have continuity at a)-to illustrate this graphically. Be sure to touch on one-sided continuity, to0. Be prepared to show/explain, graphically, what, for example, f is continuous at x=-3 f is continuous only from the right at x=0 f is discontinuous at x=2 means. (2) Now show how to go about proving/disproving continuity at a algebraically. That is, suppose you are given a function f. You want to know if f is continuous at a, without pictures. What's the procedure? Explain it out. Be sure to give an example of a piecewise function, also. For example, suppose f is the function V4x² +1 x<0 f (x)= 0. x=0. Is/f continuous at 0? If so, why? If not, why not? x* +1 x>0 (3)(Based on all this, suppose you have the function f given by ta'x? +3 f(x) = { a+9 x = 2 .2 ax? x > 2 You are given that f is continuous at x=2. Find all possible values for a. (4) Suppose you have the function f given by 4 x* - 1 x<1 f(x)= 5x³ - 9x2 +7 x21 3 Is f continuous at 1? If so, why? If not, why not?
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