How to show that this exponential function has no JUMP discontinuities?

Question

$$f(x)=a_1e^{b_1x}+a_2e^{b_2x}+\cdots+a_ne^{b_nx}$$

$$g(x)=c_1e^{d_1x}+c_2e^{d_2x}+\cdots+c_ne^{d_nx}$$

$a_n$, $b_n$, $c_n$, $d_n$ are all REAL constants and $x$ is a REAL variable

How can I show that $\frac{f(x)}{g(x)}$ cannot have any JUMP discontinuities?

Answer 1

You can write $$ \begin{gathered} \frac{{f(x)}} {{g(x)}} = \frac{{a_{\,1} e^{\,b_{\,1} \,x} }} {{c_{\,1} e^{\,d_{\,1} \,x} + \; \cdots \; + c_{\,n} e^{\,d_{\,n} \,x} }} + \; \cdots \; + \frac{{a_{\,n} e^{\,b_{\,n} \,x} }} {{c_{\,1} e^{\,d_{\,1} \,x} + \; \cdots \; + c_{\,n} e^{\,d_{\,n} \,x} }} = \hfill \\ = \frac{1} {{\left( {c_{\,1} /a_{\,1} } \right)e^{\,\left( {d_{\,1} - b_{\,1} } \right)\,x} + \; \cdots \; + \left( {c_{\,n} /a_{\,1} } \right)e^{\,\left( {d_{\,n} - b_{\,1} } \right)\,x} }} + \; \cdots \hfill \\ \end{gathered} $$ So, since $g(x)$ in general is not null and does not have discontinuities, and $1/x$ is continuous for $x \ne 0$, then ...

Answer 2

You're question is framed in the negative and it's often difficult to prove a negative.

Proving that $f/g$ is continuous is easier than proving it has NO JUMPS because if you can show it is continuous at all points then it is impossible for it to have JUMP, REMOVABLE, POINT or INFINITE discontinuities.

if $f$ and $g$ are individually continuous then their ration $f/g$ is continuous as long as $g$ is not zero. A sum of exp functions is clearly continuous since $e^{kx}$ is always continuous.

I hope that helps.

Answer 3

Erm....not quite true!!!. Try this!!

$a=arctan(1)$ and $b=arctan(2)$
define
$a(x)=-e^{2x}+8e^{x}-17+8e^{-x}-e^{-2x}$
$b(x)=e^{2x}-8e^{x}+18-8e^{-2x}+e^{-2x}$
$c^2(x)=e^{4x}-16e^{3x}+100e^{2x}-304e^{x}+454-304e^{-x}+100e^{-2x}-16e^{-3x}+e^{-4x}$ $d^2(x)=e^{4x}-16e^{3x}+98e^{2x}-288e^{x}+419-288e^{-x}+98e^{-2x}-16e^{-3x}+e^{-4x}$

then $[a(x)/b(x)]*[c(x/d(x)]*cosec^{-1}(ax/b)$

has 4 jump discontinuities (finite types) at approx +/- 1.56679923697 and +/- 0.962423650119. Where the lims differ but you'll need a darn high precision computer to verify it numerically. By jump I mean lims differ from below and above as per definition.