continuity with limsup

Define the functions

$$H(x)=\underset{\delta >0}{\inf }\underset{|y-x|<\delta }{\sup }f(y)$$

and

$$h(x)=\underset{\delta >0}{\sup }\underset{|y-x|<\delta }{\inf }f(y)$$

Then f is continuous at x iff $h(x)=H(x)$.

Note: This is an analysis proof, so we are talking about $\mathbb{R}$ with the standard topology