Continuous Functions

Key Questions

• How can I prove that a function is continuous?

  You can prove that a function #f(x)# is continuous at #a# by verifying that #lim_{x to a}f(x)=f(a)#.

  Wataru · 1 · Oct 10 2014
• Is the function #(x^2-6x+9)/(x-3)# continuous?

  No, it is not continuous at #x=3# since it is not defined there (zero denominator).

  Wataru · · Oct 10 2014
• What are continuous functions?

  Answer:

  We may also state two alternative definitions of continuous functions, using either the sequential criterion or else using topology and open sets.

  Explanation:

  Alternative definition number 1
  Let #f: X ->Y# be a function and let #(x_n)# be a sequence in X converging to an element x in X, ie #lim(x_n)=x in X#
  Then f is continuous at x iff and only if the sequence of function values converge to the image of x undr f, ie #iff lim (f(x_n))=f(x) in Y#

  Alternative definition number 2
  Let #f: X ->Y# be a function. Then f is continuous if the inverse image maps open subsets of Y into open subsets in X.
  ie, #AAA_(open)subeY =>f^(-1)(A) # is open in X

  Trevor Ryan. · 1 · Sep 17 2015

• What are continuous functions?
• What facts about continuous functions should be proved?
• How do you use continuity to evaluate the limit #sin(x+sinx)# as x approaches pi?
• How do you find values of x for which the function #g(x) = (sin(x^20+5) )^{1/3}# is continuous?
• How do you find values of x where the function #f(x)=sqrt(x^2 - 2x)# is continuous?
• How do you use continuity to evaluate the limit sin(x+sinx)?
• Given two graphs of piecewise functions f(x) and g(x), how do you know whether f[g(x)] and g[f(x)] are continuous at 0?
• How do you find the interval notation to prove #f(x)= x/(sqrt(1-x^2))# is continuous?
• How do you use continuity to evaluate the limit #(e^(x^2) - e^(-y^2)) / (x + y)# as #(xy)# approached #(1,1)#?
• How do you show that the function #f(x)=1-sqrt(1-x^2)# is continuous on the interval [-1,1]?
• Is it possible for a function to be continuous at all points in its domain and also have a one-sided limit equal to +infinite at some point?
• At what point is #f(x) = x - [x]# discontinuous?
• How do you show the function is continuous at the given number a #f(x)=(x+2x^3)^4#, a=-1?
• Is a function differentiable at all points that it is continuous?
• How do you use the definition of continuity and the properties of limits to show that the function #f(x) = 5x^4 - 9x^3 + x - 7# is continuous at a given number a=7?
• How do you use the definition of continuity and the properties of limits to show the function is continuous #F(x)= x+sqrt(x-1)# on the interval [1, inf)?
• How do you use the definition of continuity and the properties of limits to show the function is continuous #F(x)= (x^2-8)^8# on the interval (-inf, inf)?
• How do you use the definition of continuity and the properties of limits to show that the function #f(x)=x^2 + sqrt(7-x)# is continuous at a=4?
• How do you use the epsilon-delta definition of continuity to prove #f(x) = x^2# is continuous?
• How do you use the definition of continuity and the properties of limits to show that the function #h(t)=(2t-3t^2)/(1+t^3)# is continuous at the given number a=1?
• How do you use the (analysis) definition of continuity to prove the following function #f(x) = 3x +5# is continuous for all x in R?
• By the definition of continuity, how do you show that #xsin(1/x)# is continuous at x=0?
• How do you use the definition of continuity to determine if #g(x) = x^3 / x# is continuous at x=0?
• Question #a72ca
• Is it correct to write #1^@=2pi/360^@#? Yes or no, and also why?
• Question #68e60
• Question #9387b
• Is the graph of #f (x) = (X^2 + x)/x# continuous on the interval [-4, 4]?
• For what value of the constants A and B is the function (ax+3) continuous for all X if x > 5?
• For what value of the constants A and B is the function f(x)=8 continuous for all X if x = 5?
• For what value of the constants A and B is the function #f(x)=x^2+bx+1# continuous for all X if x < 5?
• If g(x)=x if x<0, x^2 if #0<= x <=1#, x^3 if x>1, how do you show that g is continuous on (all real #'s)?
• What are the three conditions necessary for the function f(x) to be continuous at the point x = c?
• What are the the three things a continuous function can't have?
• How do you find the value of k where the 2 function continuous #g(x)= |x^3|# for #-oo<x<k# and #x^2# for #k<x<oo#?
• How do you find the values of 'a' and 'b' that make f(x) continuous everywhere given #(x^2 - 4)/(x - 2)# if x < 2, #ax^2 - bx + 3# if 2 < x < 3 and #2x - a + b# if #x>=3#?
• For what value(s) of k is the function f(x) continuous at x = -3 given #f(x) = -6x - 12# when x < -3, #f(x) = k^2 - 5k# when x = -3 and f(x) = 6 when x > -3?
• If f(x) is continuous and 0 <= f(x) <= 1 for all x on the interval [0,1], then is it true that for some number x, f(x) = x?
• Why the function is discontinuous at a=1 given #f(x)= 1-x^2# if x < 1 and #f(x)= 1/x# if #x >= 1#?
• How do you find the intervals on which the function is continuous given #y = (2)/((x + 4)^2) + 8#?
• How do you find the intervals on which the function is continuous given # y = sqrt(5x + 9)#?
• How do you find the intervals on which the function is continuous given # y = ln(3x-1)#?
• Is the function f(x) = secx continuous on the interval [-pi/2, pi/2]?
• If #f(x)=(x^2+6x+8)/(x+2)# if x<-2 and #kx^2#if #x>= -2#, what value of k would make the function continuous at x=-2?
• How do we find the values of k and m that makes function continue anywhere piecewise function of #(x^2) + 5# when x > 2, #m(x+3) + k# when #-1 < x <=2# and #2(x^3) + x + 7# when #x <=-1#?
• How do you find the values of c and d that make the following function continuous for all x given #f(x) = 9x# if x<1, #f(x) = cx^2+d# if #1<=x<2# and #f(x) = 3x# if #x>=2#?
• How do you find the constant a so that the function is continuous on the entire real line given #f(x)=4, #x <= -1#, ax +b, -1< x <1 and 6, #x>=1#?
• How many continuous functions f are there which satisfy the equation #(f(x))^2 = x^2# for all x?
• How do you explain why #f(x) = x^2 +1# is continuous at x = 2?
• How do you explain why #f(x) = x^2# if #x>=1# and 2x if x<1 is continuous at x = 1?
• How do you prove that the function: #T(x) = 1 / (abs(x-2)-x^2)# is continuous between [1.5,8]?
• Consider the function defined by f(x)= sinx/x if #-1 <=x < 0#, ax+b if #0 < =x <= 1# and #(1-cosx)/x# if x >1, how do you determine a and b such that f(x) is continuous on [-1,2]?
• Is the function #2x-x^2# continuous if 0 ≤ x ≤ 2?
• Is the function #2-x# continuous if 2 < x ≤ 3?
• Is the function #x-4# continuous if 3 < x < 4?
• How do you use the definition of continuity to determine weather f is continuous at #f(x)= x-4# if #x<=0# and #x^2+x-4# if x>0?
• How do you use the definition of continuity to determine weather f is continuous at #2-x# if x<1, 1 if x=1 and #x^2# if x>1?
• Judge the following is true or false If f is continuous on ( 0,1 ) then there is a c in (0,1) such that f(c) is a maximum value of f on (0,1)?
• How do I find all intersection points of #f(x)=ax^4# and #g(x)=bx^2# where a,b > 0?
• How do you show that the function #h(x)=xe^sinx# is continuous on its domain and what is the domain?
• How do you show that the function #g(x)=sqrt(x^2-9)/(x^2-2)# is continuous on its domain and what is the domain?
• What is the continuity of the composite function #f(g(x))# given #f(x)=x^2# and #g(x)=x-1#?
• What is the continuity of the composite function #f(g(x))# given #f(x)=1/sqrtx# and #g(x)=x-1#?
• What is the continuity of the composite function #f(g(x))# given #f(x)=1/(x-6)# and #g(x)=x^2+5#?
• What is the continuity of the composite function #f(g(x))# given #f(x)=sinx# and #g(x)=x^2#?
• Question #bc73c
• Question #29a12
• Over what domain is #f(x)=sin(x)# continuous ?
• Question #b1bc4
• Question #04e10
• Question #4cbca
• How do you prove that #sqrtx# is continuous?
• The function #F(x)= x^( 2/3)# is continuous everywhere. How do you find the maximum and minimum values on [-1,2]?
• Question #36d91
• Question #f7666
• How do you find the factorial of negative numbers?
• Question #11fb2
• Question #a6851
• #lim_(x->0)(tan x - x)/x^3 = # ?
• Is the following function continuous at #x=3# ?
• Suppose #f(x)# is a real-valued function, defined and continuous on #[0, 2]# with #f(0) = -1#, #f(1) = 1# and #f(2) = -1#. Which of the following statements are true?
• Question #b2764
• Given #f(x) = sint/t#. How do you show that #f(t)# has a maximum value as #t->0#?
• Find #a# and #b# so that # f(x) = { (x^2, xle1),(ax+b,x gt 1) :} # is continuous?
• How to solve this? Determine the continuity and derivability domain for #f(x)# #f:[0,oo)->RR,f(x)=|x-1|sqrtx #
• Determine all real possible s, so that #W={f|int_-1^1f(x)dx=s}# is subspace in real function that continue in #[-1,1]# ?
• Give an example which is continous everywhere but not differentiable at 3 points?
• How to prove that #f(x)=|x|# is continue at #0# ?
• Question #34fda
• Find the values of m and b that make f continuous everywhere: m = ? b = ?
• Question #59ddc
• Which values of x this function is continuous? (x non-negative real number and n->infinity)
• Show that f(x)=2a+3b is continous, where a and b are constants?