Question

Prove that composition of two continuous function is continuous.

Answer 1

Proof : $\forall x\epsilon X,\forall \epsilon >0,\exists \delta >0$

$d(x,y)<\delta$

$\Rightarrow d\left(f\right(x),f(g\left)\right)<\epsilon$ defn. of matric square

Let f.g : $X\to X$ be a continuous function.

Let $x\epsilon X$ be fixed. Let $\epsilon >0$ be fixed.

To S.T fog is continuous.

As, f is continuous, $\exists {\delta }_f>0$ s.t

$\forall >\epsilon X,d\left(g\right(x),z)<{\delta }_f$

$\Rightarrow d\left(f\right(g\left(x\right)),f(z\left)\right)<\epsilon$ __ (1)

As g is continuous, $\exists {\delta }_g>0$ s.t

$\forall y\epsilon X,d(x,y)<{\delta }_g\Rightarrow d\left(g\right(x),g(y\left)\right)<{\delta }_f$ __ (2)

Putting (1) & (2) together,

$\forall y\epsilon X.d(x,y)<{\delta }_g\Rightarrow d\left(f\right(g\left(x\right)).g(g\left(y\right)\left)\right)<\epsilon$

Hence fog is continuous