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?

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Answer 1 · 2015-10-17T03:27:07

There is no simple answer.

For #f(g(x))#:

Find #f(g(0))#

Examine the graphs to find the one-sided limits, #lim_(xrarr0^-)f(g(x))# and #lim_(xrarr0^+)f(g(x))#

If they are equal and are the same as #f(g(0))#, then #f(g(x))# is continuous at #0#

Examples

#g_1(x) = { (x-1,x<0),(x+1,x>=0) :}#
enter image source here

#f_1(x) = { (-x+2,x<0),(x^2+2,x>=0) :}#
enter image source here

#f_1(g_1(x))# is continuous at #0#

But for #f_2(x)# shown below:
enter image source here

#f_2(g_1(x))# is not continuous at #0#.

Here is another function I'll call #g_2#

#g_2(x) = { (-x+1,x<=0),(1/x,x>0) :}#
enter image source here

Neither #f_1(g_2(x))# nor #f_1(g_2(x))# is continuous at #0#, but for #f_3# below, we have #f_3(g_2(x))# is continuous at #0#.

enter image source here

#f_3(x) = { (x^2,x<=1),(1+1/x,x>1) :}#

Perhaps someone else can summarize a method for this.