Question

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?

Answer 1

There is no simple answer.

Explanation:

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) :}`

`f_1(x) = { (-x+2,x<0),(x^2+2,x>=0) :}`

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

But for `f_2(x)` shown below:

`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) :}`

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`.

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

Perhaps someone else can summarize a method for this.