State the Lebesgue dominated convergence theorem. Find 

Any Lebesgue intergrable function is always Riemann integrable. 

Define and give an example for each of the following concepts in the context of signals and systems:

i) A stable system

ii) A time-varying system

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 . 

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 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.. 

Define saddle points. Compute the saddle points of the function  given by.