Let A and B be non-empty disjoint closed subsets of a metric space  Show that there exist open sets U ⊃ A and V ⊃ Bsuch that U ∩ V = φ.. 

Consider the function f :  → given by  sin z.

The image of any connected set in  under the functiongiven by is connected. 

Any Lebesgue intergrable function is always Riemann integrable. 

Define components in a metric space. What are all the components of the set of all nonzero real numbers under the 

1) usual metric on R, and

2) the discrete metric on R?

Let f,g ∈L′(R), define convolution f *g of f and g. Show that if either f or g is bounded, then the convolution f *g exists for all x in R and is bounded in R . 

Find the directional derivation of the function  defined by
 
 at in the direction~ .