# Thread: Analysis Continuous Functions

1. ## Analysis Continuous Functions

Let $\displaystyle f,g:R\rightarrow R$ be continuous functions. Prove directly from the definition of continuity that the function f + 5g is continuous.

2. Originally Posted by Jason Bourne
Let $\displaystyle f,g:R\rightarrow R$ be continuous functions. Prove directly from the definition of continuity that the function f + 5g is continuous.
if g is cont., then 5g is cont. so let us prove the case f + h is cont., where h = 5g
let $\displaystyle \epsilon > 0$

f,g continuous, say at x=a $\displaystyle \implies \exists \delta _1 , \delta _2$ such that if

i) $\displaystyle |x-a| < \delta_1 \implies |f(x) - f(a)| < \frac{\epsilon }{2}$

ii) $\displaystyle |x-a| < \delta_2 \implies |h(x) - h(a)| < \frac{\epsilon }{2}$

take $\displaystyle \delta = min \{ \delta_1 , \delta_2 \}$

then if $\displaystyle |x - a| < \delta$,

$\displaystyle |(f+h)(x) - (f+h)(a)| = |f(x) - f(a) + h(x) - h(a)|$

$\displaystyle \leq |f(x) - f(a)| + |h(x) - h(a)| < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon$. QED