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 . 

State the Lebesgue dominated convergence theorem. Find 

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

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

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 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.. 

The functiongiven by  is differentiable at 0.