# continuity - analysis

• December 5th 2007, 01:02 PM
Jason Bourne
continuity - analysis
Let $f,g: \Re \rightarrow \Re$ be continuous functions. Prove directly from the definition of continuity that the function $\varphi : \Re \rightarrow \Re$ defined by

$\varphi (x) = 7f(x) - 3g(x) + 5x$

is continuous
• December 5th 2007, 01:38 PM
Plato
This is real tedious problem. Start with some point a.
The given $\varepsilon > 0$ then for the functions $f(x)\,,\,g(x)\,\& \,x$
find three deltas such that:
$\begin{array}{l} 0 < \left| {x - a} \right| < \delta _1 \quad \Rightarrow \quad \left| {f(x) - f(a)} \right| < \frac{\varepsilon }{{21}} \\
0 < \left| {x - a} \right| < \delta _2 \quad \Rightarrow \quad \left| {g(x) - g(a)} \right| < \frac{\varepsilon }{9} \\
0 < \left| {x - a} \right| < \delta _3 \quad \Rightarrow \quad \left| {x - a} \right| < \frac{\varepsilon }{{15}} \\
\end{array}$